Special¶

DiracDelta¶

class sympy.functions.special.delta_functions.DiracDelta[source]¶

The DiracDelta function and its derivatives.

DiracDelta is not an ordinary function. It can be rigorously defined either as a distribution or as a measure.

DiracDelta only makes sense in definite integrals, and in particular, integrals of the form Integral(f(x)*DiracDelta(x - x0), (x, a, b)), where it equals f(x0) if a <= x0 <= b and 0 otherwise. Formally, DiracDelta acts in some ways like a function that is 0 everywhere except at 0, but in many ways it also does not. It can often be useful to treat DiracDelta in formal ways, building up and manipulating expressions with delta functions (which may eventually be integrated), but care must be taken to not treat it as a real function. SymPy’s oo is similar. It only truly makes sense formally in certain contexts (such as integration limits), but SymPy allows its use everywhere, and it tries to be consistent with operations on it (like 1/oo), but it is easy to get into trouble and get wrong results if oo is treated too much like a number. Similarly, if DiracDelta is treated too much like a function, it is easy to get wrong or nonsensical results.

DiracDelta function has the following properties:

diff(Heaviside(x), x) = DiracDelta(x)
integrate(DiracDelta(x - a)*f(x),(x, -oo, oo)) = f(a) and integrate(DiracDelta(x - a)*f(x),(x, a - e, a + e)) = f(a)
DiracDelta(x) = 0 for all x != 0
DiracDelta(g(x)) = Sum_i(DiracDelta(x - x_i)/abs(g'(x_i))) Where x_i-s are the roots of g
DiracDelta(-x) = DiracDelta(x)

Derivatives of k-th order of DiracDelta have the following property:

DiracDelta(x, k) = 0, for all x != 0
DiracDelta(-x, k) = -DiracDelta(x, k) for odd k
DiracDelta(-x, k) = DiracDelta(x, k) for even k

Examples

>>> from sympy import DiracDelta, diff, pi, Piecewise
>>> from sympy.abc import x, y

>>> DiracDelta(x)
DiracDelta(x)
>>> DiracDelta(1)
0
>>> DiracDelta(-1)
0
>>> DiracDelta(pi)
0
>>> DiracDelta(x - 4).subs(x, 4)
DiracDelta(0)
>>> diff(DiracDelta(x))
DiracDelta(x, 1)
>>> diff(DiracDelta(x - 1),x,2)
DiracDelta(x - 1, 2)
>>> diff(DiracDelta(x**2 - 1),x,2)
2*(2*x**2*DiracDelta(x**2 - 1, 2) + DiracDelta(x**2 - 1, 1))
>>> DiracDelta(3*x).is_simple(x)
True
>>> DiracDelta(x**2).is_simple(x)
False
>>> DiracDelta((x**2 - 1)*y).expand(diracdelta=True, wrt=x)
DiracDelta(x - 1)/(2*Abs(y)) + DiracDelta(x + 1)/(2*Abs(y))

References

R236: http://mathworld.wolfram.com/DeltaFunction.html

classmethod eval(arg, k=0)[source]¶

Returns a simplified form or a value of DiracDelta depending on the argument passed by the DiracDelta object.

The eval() method is automatically called when the DiracDelta class is about to be instantiated and it returns either some simplified instance or the unevaluated instance depending on the argument passed. In other words, eval() method is not needed to be called explicitly, it is being called and evaluated once the object is called.

Examples

>>> from sympy import DiracDelta, S, Subs
>>> from sympy.abc import x

>>> DiracDelta(x)
DiracDelta(x)

>>> DiracDelta(-x, 1)
-DiracDelta(x, 1)

>>> DiracDelta(1)
0

>>> DiracDelta(5, 1)
0

>>> DiracDelta(0)
DiracDelta(0)

>>> DiracDelta(-1)
0

>>> DiracDelta(S.NaN)
nan

>>> DiracDelta(x).eval(1)
0

>>> DiracDelta(x - 100).subs(x, 5)
0

>>> DiracDelta(x - 100).subs(x, 100)
DiracDelta(0)

fdiff(argindex=1)[source]¶

Returns the first derivative of a DiracDelta Function.

The difference between diff() and fdiff() is:- diff() is the user-level function and fdiff() is an object method. fdiff() is just a convenience method available in the Function class. It returns the derivative of the function without considering the chain rule. diff(function, x) calls Function._eval_derivative which in turn calls fdiff() internally to compute the derivative of the function.

Examples

>>> from sympy import DiracDelta, diff
>>> from sympy.abc import x

>>> DiracDelta(x).fdiff()
DiracDelta(x, 1)

>>> DiracDelta(x, 1).fdiff()
DiracDelta(x, 2)

>>> DiracDelta(x**2 - 1).fdiff()
DiracDelta(x**2 - 1, 1)

>>> diff(DiracDelta(x, 1)).fdiff()
DiracDelta(x, 3)

is_simple(self, x)[source]¶

Tells whether the argument(args[0]) of DiracDelta is a linear expression in x.

x can be:

a symbol

Examples

>>> from sympy import DiracDelta, cos
>>> from sympy.abc import x, y

>>> DiracDelta(x*y).is_simple(x)
True
>>> DiracDelta(x*y).is_simple(y)
True

>>> DiracDelta(x**2 + x - 2).is_simple(x)
False

>>> DiracDelta(cos(x)).is_simple(x)
False

See also

simplify, Diracdelta

Heaviside¶

class sympy.functions.special.delta_functions.Heaviside[source]¶

Heaviside Piecewise function

Heaviside function has the following properties [R237]:

diff(Heaviside(x),x) = DiracDelta(x)
( 0, if x < 0
Heaviside(x) = < ( undefined if x==0 [1]
( 1, if x > 0
Max(0,x).diff(x) = Heaviside(x)

R237: Regarding to the value at 0, Mathematica defines H(0) = 1, but Maple uses H(0) = undefined. Different application areas may have specific conventions. For example, in control theory, it is common practice to assume H(0) == 0 to match the Laplace transform of a DiracDelta distribution.

To specify the value of Heaviside at x=0, a second argument can be given. Omit this 2nd argument or pass None to recover the default behavior.

>>> from sympy import Heaviside, S
>>> from sympy.abc import x
>>> Heaviside(9)
1
>>> Heaviside(-9)
0
>>> Heaviside(0)
Heaviside(0)
>>> Heaviside(0, S.Half)
1/2
>>> (Heaviside(x) + 1).replace(Heaviside(x), Heaviside(x, 1))
Heaviside(x, 1) + 1

See also

DiracDelta

References

R238: http://mathworld.wolfram.com/HeavisideStepFunction.html
R239: http://dlmf.nist.gov/1.16#iv

classmethod eval(arg, H0=None)[source]¶

Returns a simplified form or a value of Heaviside depending on the argument passed by the Heaviside object.

The eval() method is automatically called when the Heaviside class is about to be instantiated and it returns either some simplified instance or the unevaluated instance depending on the argument passed. In other words, eval() method is not needed to be called explicitly, it is being called and evaluated once the object is called.

Examples

>>> from sympy import Heaviside, S
>>> from sympy.abc import x

>>> Heaviside(x)
Heaviside(x)

>>> Heaviside(19)
1

>>> Heaviside(0)
Heaviside(0)

>>> Heaviside(0, 1)
1

>>> Heaviside(-5)
0

>>> Heaviside(S.NaN)
nan

>>> Heaviside(x).eval(100)
1

>>> Heaviside(x - 100).subs(x, 5)
0

>>> Heaviside(x - 100).subs(x, 105)
1

fdiff(argindex=1)[source]¶

Returns the first derivative of a Heaviside Function.

Examples

>>> from sympy import Heaviside, diff
>>> from sympy.abc import x

>>> Heaviside(x).fdiff()
DiracDelta(x)

>>> Heaviside(x**2 - 1).fdiff()
DiracDelta(x**2 - 1)

>>> diff(Heaviside(x)).fdiff()
DiracDelta(x, 1)

Singularity Function¶

class sympy.functions.special.singularity_functions.SingularityFunction[source]¶

The Singularity functions are a class of discontinuous functions. They take a variable, an offset and an exponent as arguments. These functions are represented using Macaulay brackets as :

SingularityFunction(x, a, n) := <x - a>^n

The singularity function will automatically evaluate to Derivative(DiracDelta(x - a), x, -n - 1) if n < 0 and (x - a)**n*Heaviside(x - a) if n >= 0.

Examples

>>> from sympy import SingularityFunction, diff, Piecewise, DiracDelta, Heaviside, Symbol
>>> from sympy.abc import x, a, n
>>> SingularityFunction(x, a, n)
SingularityFunction(x, a, n)
>>> y = Symbol('y', positive=True)
>>> n = Symbol('n', nonnegative=True)
>>> SingularityFunction(y, -10, n)
(y + 10)**n
>>> y = Symbol('y', negative=True)
>>> SingularityFunction(y, 10, n)
0
>>> SingularityFunction(x, 4, -1).subs(x, 4)
oo
>>> SingularityFunction(x, 10, -2).subs(x, 10)
oo
>>> SingularityFunction(4, 1, 5)
243
>>> diff(SingularityFunction(x, 1, 5) + SingularityFunction(x, 1, 4), x)
4*SingularityFunction(x, 1, 3) + 5*SingularityFunction(x, 1, 4)
>>> diff(SingularityFunction(x, 4, 0), x, 2)
SingularityFunction(x, 4, -2)
>>> SingularityFunction(x, 4, 5).rewrite(Piecewise)
Piecewise(((x - 4)**5, x - 4 > 0), (0, True))
>>> expr = SingularityFunction(x, a, n)
>>> y = Symbol('y', positive=True)
>>> n = Symbol('n', nonnegative=True)
>>> expr.subs({x: y, a: -10, n: n})
(y + 10)**n

The methods rewrite(DiracDelta), rewrite(Heaviside) and rewrite('HeavisideDiracDelta') returns the same output. One can use any of these methods according to their choice.

>>> expr = SingularityFunction(x, 4, 5) + SingularityFunction(x, -3, -1) - SingularityFunction(x, 0, -2)
>>> expr.rewrite(Heaviside)
(x - 4)**5*Heaviside(x - 4) + DiracDelta(x + 3) - DiracDelta(x, 1)
>>> expr.rewrite(DiracDelta)
(x - 4)**5*Heaviside(x - 4) + DiracDelta(x + 3) - DiracDelta(x, 1)
>>> expr.rewrite('HeavisideDiracDelta')
(x - 4)**5*Heaviside(x - 4) + DiracDelta(x + 3) - DiracDelta(x, 1)

Reference

R240: https://en.wikipedia.org/wiki/Singularity_function

See also

DiracDelta, Heaviside

classmethod eval(variable, offset, exponent)[source]¶

Returns a simplified form or a value of Singularity Function depending on the argument passed by the object.

The eval() method is automatically called when the SingularityFunction class is about to be instantiated and it returns either some simplified instance or the unevaluated instance depending on the argument passed. In other words, eval() method is not needed to be called explicitly, it is being called and evaluated once the object is called.

Examples

>>> from sympy import SingularityFunction, Symbol, nan
>>> from sympy.abc import x, a, n
>>> SingularityFunction(x, a, n)
SingularityFunction(x, a, n)
>>> SingularityFunction(5, 3, 2)
4
>>> SingularityFunction(x, a, nan)
nan
>>> SingularityFunction(x, 3, 0).subs(x, 3)
1
>>> SingularityFunction(x, a, n).eval(3, 5, 1)
0
>>> SingularityFunction(x, a, n).eval(4, 1, 5)
243
>>> x = Symbol('x', positive = True)
>>> a = Symbol('a', negative = True)
>>> n = Symbol('n', nonnegative = True)
>>> SingularityFunction(x, a, n)
(-a + x)**n
>>> x = Symbol('x', negative = True)
>>> a = Symbol('a', positive = True)
>>> SingularityFunction(x, a, n)
0

fdiff(argindex=1)[source]¶

Returns the first derivative of a DiracDelta Function.

The difference between diff() and fdiff() is:- diff() is the user-level function and fdiff() is an object method. fdiff() is just a convenience method available in the Function class. It returns the derivative of the function without considering the chain rule. diff(function, x) calls Function._eval_derivative which in turn calls fdiff() internally to compute the derivative of the function.

Gamma, Beta and related Functions¶

class sympy.functions.special.gamma_functions.gamma[source]¶

The gamma function

\[\Gamma(x) := \int^{\infty}_{0} t^{x-1} e^{-t} \mathrm{d}t.\]

The gamma function implements the function which passes through the values of the factorial function, i.e. \(\Gamma(n) = (n - 1)!\) when n is an integer. More general, \(\Gamma(z)\) is defined in the whole complex plane except at the negative integers where there are simple poles.

Examples

>>> from sympy import S, I, pi, oo, gamma
>>> from sympy.abc import x

Several special values are known:

>>> gamma(1)
1
>>> gamma(4)
6
>>> gamma(S(3)/2)
sqrt(pi)/2

The Gamma function obeys the mirror symmetry:

>>> from sympy import conjugate
>>> conjugate(gamma(x))
gamma(conjugate(x))

Differentiation with respect to x is supported:

>>> from sympy import diff
>>> diff(gamma(x), x)
gamma(x)*polygamma(0, x)

Series expansion is also supported:

>>> from sympy import series
>>> series(gamma(x), x, 0, 3)
1/x - EulerGamma + x*(EulerGamma**2/2 + pi**2/12) + x**2*(-EulerGamma*pi**2/12 + polygamma(2, 1)/6 - EulerGamma**3/6) + O(x**3)

We can numerically evaluate the gamma function to arbitrary precision on the whole complex plane:

>>> gamma(pi).evalf(40)
2.288037795340032417959588909060233922890
>>> gamma(1+I).evalf(20)
0.49801566811835604271 - 0.15494982830181068512*I

See also

lowergamma: Lower incomplete gamma function.
uppergamma: Upper incomplete gamma function.
polygamma: Polygamma function.
loggamma: Log Gamma function.
digamma: Digamma function.
trigamma: Trigamma function.
sympy.functions.special.beta_functions.beta: Euler Beta function.

References

R241: https://en.wikipedia.org/wiki/Gamma_function
R242: http://dlmf.nist.gov/5
R243: http://mathworld.wolfram.com/GammaFunction.html
R244: http://functions.wolfram.com/GammaBetaErf/Gamma/

class sympy.functions.special.gamma_functions.loggamma[source]¶

The loggamma function implements the logarithm of the gamma function i.e, \(\log\Gamma(x)\).

Examples

Several special values are known. For numerical integral arguments we have:

>>> from sympy import loggamma
>>> loggamma(-2)
oo
>>> loggamma(0)
oo
>>> loggamma(1)
0
>>> loggamma(2)
0
>>> loggamma(3)
log(2)

and for symbolic values:

>>> from sympy import Symbol
>>> n = Symbol("n", integer=True, positive=True)
>>> loggamma(n)
log(gamma(n))
>>> loggamma(-n)
oo

for half-integral values:

>>> from sympy import S, pi
>>> loggamma(S(5)/2)
log(3*sqrt(pi)/4)
>>> loggamma(n/2)
log(2**(1 - n)*sqrt(pi)*gamma(n)/gamma(n/2 + 1/2))

and general rational arguments:

>>> from sympy import expand_func
>>> L = loggamma(S(16)/3)
>>> expand_func(L).doit()
-5*log(3) + loggamma(1/3) + log(4) + log(7) + log(10) + log(13)
>>> L = loggamma(S(19)/4)
>>> expand_func(L).doit()
-4*log(4) + loggamma(3/4) + log(3) + log(7) + log(11) + log(15)
>>> L = loggamma(S(23)/7)
>>> expand_func(L).doit()
-3*log(7) + log(2) + loggamma(2/7) + log(9) + log(16)

The loggamma function has the following limits towards infinity:

>>> from sympy import oo
>>> loggamma(oo)
oo
>>> loggamma(-oo)
zoo

The loggamma function obeys the mirror symmetry if \(x \in \mathbb{C} \setminus \{-\infty, 0\}\):

>>> from sympy.abc import x
>>> from sympy import conjugate
>>> conjugate(loggamma(x))
loggamma(conjugate(x))

Differentiation with respect to x is supported:

>>> from sympy import diff
>>> diff(loggamma(x), x)
polygamma(0, x)

Series expansion is also supported:

>>> from sympy import series
>>> series(loggamma(x), x, 0, 4)
-log(x) - EulerGamma*x + pi**2*x**2/12 + x**3*polygamma(2, 1)/6 + O(x**4)

We can numerically evaluate the gamma function to arbitrary precision on the whole complex plane:

>>> from sympy import I
>>> loggamma(5).evalf(30)
3.17805383034794561964694160130
>>> loggamma(I).evalf(20)
-0.65092319930185633889 - 1.8724366472624298171*I

See also

gamma: Gamma function.
lowergamma: Lower incomplete gamma function.
uppergamma: Upper incomplete gamma function.
polygamma: Polygamma function.
digamma: Digamma function.
trigamma: Trigamma function.
sympy.functions.special.beta_functions.beta: Euler Beta function.

References

R245: https://en.wikipedia.org/wiki/Gamma_function
R246: http://dlmf.nist.gov/5
R247: http://mathworld.wolfram.com/LogGammaFunction.html
R248: http://functions.wolfram.com/GammaBetaErf/LogGamma/

class sympy.functions.special.gamma_functions.polygamma[source]¶

The function polygamma(n, z) returns log(gamma(z)).diff(n + 1).

It is a meromorphic function on \(\mathbb{C}\) and defined as the (n+1)-th derivative of the logarithm of the gamma function:

\[\psi^{(n)} (z) := \frac{\mathrm{d}^{n+1}}{\mathrm{d} z^{n+1}} \log\Gamma(z).\]

Examples

Several special values are known:

>>> from sympy import S, polygamma
>>> polygamma(0, 1)
-EulerGamma
>>> polygamma(0, 1/S(2))
-2*log(2) - EulerGamma
>>> polygamma(0, 1/S(3))
-log(3) - sqrt(3)*pi/6 - EulerGamma - log(sqrt(3))
>>> polygamma(0, 1/S(4))
-pi/2 - log(4) - log(2) - EulerGamma
>>> polygamma(0, 2)
1 - EulerGamma
>>> polygamma(0, 23)
19093197/5173168 - EulerGamma

>>> from sympy import oo, I
>>> polygamma(0, oo)
oo
>>> polygamma(0, -oo)
oo
>>> polygamma(0, I*oo)
oo
>>> polygamma(0, -I*oo)
oo

Differentiation with respect to x is supported:

>>> from sympy import Symbol, diff
>>> x = Symbol("x")
>>> diff(polygamma(0, x), x)
polygamma(1, x)
>>> diff(polygamma(0, x), x, 2)
polygamma(2, x)
>>> diff(polygamma(0, x), x, 3)
polygamma(3, x)
>>> diff(polygamma(1, x), x)
polygamma(2, x)
>>> diff(polygamma(1, x), x, 2)
polygamma(3, x)
>>> diff(polygamma(2, x), x)
polygamma(3, x)
>>> diff(polygamma(2, x), x, 2)
polygamma(4, x)

>>> n = Symbol("n")
>>> diff(polygamma(n, x), x)
polygamma(n + 1, x)
>>> diff(polygamma(n, x), x, 2)
polygamma(n + 2, x)

We can rewrite polygamma functions in terms of harmonic numbers:

>>> from sympy import harmonic
>>> polygamma(0, x).rewrite(harmonic)
harmonic(x - 1) - EulerGamma
>>> polygamma(2, x).rewrite(harmonic)
2*harmonic(x - 1, 3) - 2*zeta(3)
>>> ni = Symbol("n", integer=True)
>>> polygamma(ni, x).rewrite(harmonic)
(-1)**(n + 1)*(-harmonic(x - 1, n + 1) + zeta(n + 1))*factorial(n)

See also

gamma: Gamma function.
lowergamma: Lower incomplete gamma function.
uppergamma: Upper incomplete gamma function.
loggamma: Log Gamma function.
digamma: Digamma function.
trigamma: Trigamma function.
sympy.functions.special.beta_functions.beta: Euler Beta function.

References

R249: https://en.wikipedia.org/wiki/Polygamma_function
R250: http://mathworld.wolfram.com/PolygammaFunction.html
R251: http://functions.wolfram.com/GammaBetaErf/PolyGamma/
R252: http://functions.wolfram.com/GammaBetaErf/PolyGamma2/

sympy.functions.special.gamma_functions.digamma(x)[source]¶

The digamma function is the first derivative of the loggamma function i.e,

\[\psi(x) := \frac{\mathrm{d}}{\mathrm{d} z} \log\Gamma(z) = \frac{\Gamma'(z)}{\Gamma(z) }\]

In this case, digamma(z) = polygamma(0, z).

See also

gamma: Gamma function.
lowergamma: Lower incomplete gamma function.
uppergamma: Upper incomplete gamma function.
polygamma: Polygamma function.
loggamma: Log Gamma function.
trigamma: Trigamma function.
sympy.functions.special.beta_functions.beta: Euler Beta function.

References

R253: https://en.wikipedia.org/wiki/Digamma_function
R254: http://mathworld.wolfram.com/DigammaFunction.html
R255: http://functions.wolfram.com/GammaBetaErf/PolyGamma2/

sympy.functions.special.gamma_functions.trigamma(x)[source]¶

The trigamma function is the second derivative of the loggamma function i.e,

\[\psi^{(1)}(z) := \frac{\mathrm{d}^{2}}{\mathrm{d} z^{2}} \log\Gamma(z).\]

In this case, trigamma(z) = polygamma(1, z).

See also

gamma: Gamma function.
lowergamma: Lower incomplete gamma function.
uppergamma: Upper incomplete gamma function.
polygamma: Polygamma function.
loggamma: Log Gamma function.
digamma: Digamma function.
sympy.functions.special.beta_functions.beta: Euler Beta function.

References

R256: https://en.wikipedia.org/wiki/Trigamma_function
R257: http://mathworld.wolfram.com/TrigammaFunction.html
R258: http://functions.wolfram.com/GammaBetaErf/PolyGamma2/

class sympy.functions.special.gamma_functions.uppergamma[source]¶

The upper incomplete gamma function.

It can be defined as the meromorphic continuation of

\[\Gamma(s, x) := \int_x^\infty t^{s-1} e^{-t} \mathrm{d}t = \Gamma(s) - \gamma(s, x).\]

where \(\gamma(s, x)\) is the lower incomplete gamma function, lowergamma. This can be shown to be the same as

\[\Gamma(s, x) = \Gamma(s) - \frac{x^s}{s} {}_1F_1\left({s \atop s+1} \middle| -x\right),\]

where \({}_1F_1\) is the (confluent) hypergeometric function.

The upper incomplete gamma function is also essentially equivalent to the generalized exponential integral:

\[\operatorname{E}_{n}(x) = \int_{1}^{\infty}{\frac{e^{-xt}}{t^n} \, dt} = x^{n-1}\Gamma(1-n,x).\]

Examples

>>> from sympy import uppergamma, S
>>> from sympy.abc import s, x
>>> uppergamma(s, x)
uppergamma(s, x)
>>> uppergamma(3, x)
2*(x**2/2 + x + 1)*exp(-x)
>>> uppergamma(-S(1)/2, x)
-2*sqrt(pi)*erfc(sqrt(x)) + 2*exp(-x)/sqrt(x)
>>> uppergamma(-2, x)
expint(3, x)/x**2

See also

gamma: Gamma function.
lowergamma: Lower incomplete gamma function.
polygamma: Polygamma function.
loggamma: Log Gamma function.
digamma: Digamma function.
trigamma: Trigamma function.
sympy.functions.special.beta_functions.beta: Euler Beta function.

References

R259: https://en.wikipedia.org/wiki/Incomplete_gamma_function#Upper_incomplete_Gamma_function
R260: Abramowitz, Milton; Stegun, Irene A., eds. (1965), Chapter 6, Section 5, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
R261: http://dlmf.nist.gov/8
R262: http://functions.wolfram.com/GammaBetaErf/Gamma2/
R263: http://functions.wolfram.com/GammaBetaErf/Gamma3/
R264: https://en.wikipedia.org/wiki/Exponential_integral#Relation_with_other_functions

class sympy.functions.special.gamma_functions.lowergamma[source]¶

The lower incomplete gamma function.

It can be defined as the meromorphic continuation of

\[\gamma(s, x) := \int_0^x t^{s-1} e^{-t} \mathrm{d}t = \Gamma(s) - \Gamma(s, x).\]

This can be shown to be the same as

\[\gamma(s, x) = \frac{x^s}{s} {}_1F_1\left({s \atop s+1} \middle| -x\right),\]

where \({}_1F_1\) is the (confluent) hypergeometric function.

Examples

>>> from sympy import lowergamma, S
>>> from sympy.abc import s, x
>>> lowergamma(s, x)
lowergamma(s, x)
>>> lowergamma(3, x)
-2*(x**2/2 + x + 1)*exp(-x) + 2
>>> lowergamma(-S(1)/2, x)
-2*sqrt(pi)*erf(sqrt(x)) - 2*exp(-x)/sqrt(x)

See also

gamma: Gamma function.
uppergamma: Upper incomplete gamma function.
polygamma: Polygamma function.
loggamma: Log Gamma function.
digamma: Digamma function.
trigamma: Trigamma function.
sympy.functions.special.beta_functions.beta: Euler Beta function.

References

R265: https://en.wikipedia.org/wiki/Incomplete_gamma_function#Lower_incomplete_Gamma_function
R266: Abramowitz, Milton; Stegun, Irene A., eds. (1965), Chapter 6, Section 5, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
R267: http://dlmf.nist.gov/8
R268: http://functions.wolfram.com/GammaBetaErf/Gamma2/
R269: http://functions.wolfram.com/GammaBetaErf/Gamma3/

class sympy.functions.special.beta_functions.beta[source]¶

The beta integral is called the Eulerian integral of the first kind by Legendre:

\[\mathrm{B}(x,y) := \int^{1}_{0} t^{x-1} (1-t)^{y-1} \mathrm{d}t.\]

Beta function or Euler’s first integral is closely associated with gamma function. The Beta function often used in probability theory and mathematical statistics. It satisfies properties like:

\[\begin{split}\mathrm{B}(a,1) = \frac{1}{a} \\ \mathrm{B}(a,b) = \mathrm{B}(b,a) \\ \mathrm{B}(a,b) = \frac{\Gamma(a) \Gamma(b)}{\Gamma(a+b)}\end{split}\]

Therefore for integral values of a and b:

\[\mathrm{B} = \frac{(a-1)! (b-1)!}{(a+b-1)!}\]

Examples

>>> from sympy import I, pi
>>> from sympy.abc import x,y

The Beta function obeys the mirror symmetry:

>>> from sympy import beta
>>> from sympy import conjugate
>>> conjugate(beta(x,y))
beta(conjugate(x), conjugate(y))

Differentiation with respect to both x and y is supported:

>>> from sympy import beta
>>> from sympy import diff
>>> diff(beta(x,y), x)
(polygamma(0, x) - polygamma(0, x + y))*beta(x, y)

>>> from sympy import beta
>>> from sympy import diff
>>> diff(beta(x,y), y)
(polygamma(0, y) - polygamma(0, x + y))*beta(x, y)

We can numerically evaluate the gamma function to arbitrary precision on the whole complex plane:

>>> from sympy import beta
>>> beta(pi,pi).evalf(40)
0.02671848900111377452242355235388489324562

>>> beta(1+I,1+I).evalf(20)
-0.2112723729365330143 - 0.7655283165378005676*I

See also

sympy.functions.special.gamma_functions.gamma: Gamma function.
sympy.functions.special.gamma_functions.uppergamma: Upper incomplete gamma function.
sympy.functions.special.gamma_functions.lowergamma: Lower incomplete gamma function.
sympy.functions.special.gamma_functions.polygamma: Polygamma function.
sympy.functions.special.gamma_functions.loggamma: Log Gamma function.
sympy.functions.special.gamma_functions.digamma: Digamma function.
sympy.functions.special.gamma_functions.trigamma: Trigamma function.

References

R270: https://en.wikipedia.org/wiki/Beta_function
R271: http://mathworld.wolfram.com/BetaFunction.html
R272: http://dlmf.nist.gov/5.12

Error Functions and Fresnel Integrals¶

class sympy.functions.special.error_functions.erf[source]¶

The Gauss error function. This function is defined as:

\[\mathrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \mathrm{d}t.\]

Examples

>>> from sympy import I, oo, erf
>>> from sympy.abc import z

Several special values are known:

>>> erf(0)
0
>>> erf(oo)
1
>>> erf(-oo)
-1
>>> erf(I*oo)
oo*I
>>> erf(-I*oo)
-oo*I

In general one can pull out factors of -1 and I from the argument:

>>> erf(-z)
-erf(z)

The error function obeys the mirror symmetry:

>>> from sympy import conjugate
>>> conjugate(erf(z))
erf(conjugate(z))

Differentiation with respect to z is supported:

>>> from sympy import diff
>>> diff(erf(z), z)
2*exp(-z**2)/sqrt(pi)

We can numerically evaluate the error function to arbitrary precision on the whole complex plane:

>>> erf(4).evalf(30)
0.999999984582742099719981147840

>>> erf(-4*I).evalf(30)
-1296959.73071763923152794095062*I

See also

erfc: Complementary error function.
erfi: Imaginary error function.
erf2: Two-argument error function.
erfinv: Inverse error function.
erfcinv: Inverse Complementary error function.
erf2inv: Inverse two-argument error function.

References

R273: https://en.wikipedia.org/wiki/Error_function
R274: http://dlmf.nist.gov/7
R275: http://mathworld.wolfram.com/Erf.html
R276: http://functions.wolfram.com/GammaBetaErf/Erf

class sympy.functions.special.error_functions.erfc[source]¶

Complementary Error Function. The function is defined as:

\[\mathrm{erfc}(x) = \frac{2}{\sqrt{\pi}} \int_x^\infty e^{-t^2} \mathrm{d}t\]

Examples

>>> from sympy import I, oo, erfc
>>> from sympy.abc import z

Several special values are known:

>>> erfc(0)
1
>>> erfc(oo)
0
>>> erfc(-oo)
2
>>> erfc(I*oo)
-oo*I
>>> erfc(-I*oo)
oo*I

The error function obeys the mirror symmetry:

>>> from sympy import conjugate
>>> conjugate(erfc(z))
erfc(conjugate(z))

Differentiation with respect to z is supported:

>>> from sympy import diff
>>> diff(erfc(z), z)
-2*exp(-z**2)/sqrt(pi)

It also follows

>>> erfc(-z)
2 - erfc(z)

We can numerically evaluate the complementary error function to arbitrary precision on the whole complex plane:

>>> erfc(4).evalf(30)
0.0000000154172579002800188521596734869

>>> erfc(4*I).evalf(30)
1.0 - 1296959.73071763923152794095062*I

See also

erf: Gaussian error function.
erfi: Imaginary error function.
erf2: Two-argument error function.
erfinv: Inverse error function.
erfcinv: Inverse Complementary error function.
erf2inv: Inverse two-argument error function.

References

R277: https://en.wikipedia.org/wiki/Error_function
R278: http://dlmf.nist.gov/7
R279: http://mathworld.wolfram.com/Erfc.html
R280: http://functions.wolfram.com/GammaBetaErf/Erfc

class sympy.functions.special.error_functions.erfi[source]¶

Imaginary error function. The function erfi is defined as:

\[\mathrm{erfi}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{t^2} \mathrm{d}t\]

Examples

>>> from sympy import I, oo, erfi
>>> from sympy.abc import z

Several special values are known:

>>> erfi(0)
0
>>> erfi(oo)
oo
>>> erfi(-oo)
-oo
>>> erfi(I*oo)
I
>>> erfi(-I*oo)
-I

In general one can pull out factors of -1 and I from the argument:

>>> erfi(-z)
-erfi(z)

>>> from sympy import conjugate
>>> conjugate(erfi(z))
erfi(conjugate(z))

Differentiation with respect to z is supported:

>>> from sympy import diff
>>> diff(erfi(z), z)
2*exp(z**2)/sqrt(pi)

We can numerically evaluate the imaginary error function to arbitrary precision on the whole complex plane:

>>> erfi(2).evalf(30)
18.5648024145755525987042919132

>>> erfi(-2*I).evalf(30)
-0.995322265018952734162069256367*I

See also

erf: Gaussian error function.
erfc: Complementary error function.
erf2: Two-argument error function.
erfinv: Inverse error function.
erfcinv: Inverse Complementary error function.
erf2inv: Inverse two-argument error function.

References

R281: https://en.wikipedia.org/wiki/Error_function
R282: http://mathworld.wolfram.com/Erfi.html
R283: http://functions.wolfram.com/GammaBetaErf/Erfi

class sympy.functions.special.error_functions.erf2[source]¶

Two-argument error function. This function is defined as:

\[\mathrm{erf2}(x, y) = \frac{2}{\sqrt{\pi}} \int_x^y e^{-t^2} \mathrm{d}t\]

Examples

>>> from sympy import I, oo, erf2
>>> from sympy.abc import x, y

Several special values are known:

>>> erf2(0, 0)
0
>>> erf2(x, x)
0
>>> erf2(x, oo)
1 - erf(x)
>>> erf2(x, -oo)
-erf(x) - 1
>>> erf2(oo, y)
erf(y) - 1
>>> erf2(-oo, y)
erf(y) + 1

In general one can pull out factors of -1:

>>> erf2(-x, -y)
-erf2(x, y)

The error function obeys the mirror symmetry:

>>> from sympy import conjugate
>>> conjugate(erf2(x, y))
erf2(conjugate(x), conjugate(y))

Differentiation with respect to x, y is supported:

>>> from sympy import diff
>>> diff(erf2(x, y), x)
-2*exp(-x**2)/sqrt(pi)
>>> diff(erf2(x, y), y)
2*exp(-y**2)/sqrt(pi)

See also

erf: Gaussian error function.
erfc: Complementary error function.
erfi: Imaginary error function.
erfinv: Inverse error function.
erfcinv: Inverse Complementary error function.
erf2inv: Inverse two-argument error function.

References

R284: http://functions.wolfram.com/GammaBetaErf/Erf2/

class sympy.functions.special.error_functions.erfinv[source]¶

Inverse Error Function. The erfinv function is defined as:

\[\mathrm{erf}(x) = y \quad \Rightarrow \quad \mathrm{erfinv}(y) = x\]

Examples

>>> from sympy import I, oo, erfinv
>>> from sympy.abc import x

Several special values are known:

>>> erfinv(0)
0
>>> erfinv(1)
oo

Differentiation with respect to x is supported:

>>> from sympy import diff
>>> diff(erfinv(x), x)
sqrt(pi)*exp(erfinv(x)**2)/2

We can numerically evaluate the inverse error function to arbitrary precision on [-1, 1]:

>>> erfinv(0.2).evalf(30)
0.179143454621291692285822705344

See also

erf: Gaussian error function.
erfc: Complementary error function.
erfi: Imaginary error function.
erf2: Two-argument error function.
erfcinv: Inverse Complementary error function.
erf2inv: Inverse two-argument error function.

References

R285: https://en.wikipedia.org/wiki/Error_function#Inverse_functions
R286: http://functions.wolfram.com/GammaBetaErf/InverseErf/

class sympy.functions.special.error_functions.erfcinv[source]¶

Inverse Complementary Error Function. The erfcinv function is defined as:

\[\mathrm{erfc}(x) = y \quad \Rightarrow \quad \mathrm{erfcinv}(y) = x\]

Examples

>>> from sympy import I, oo, erfcinv
>>> from sympy.abc import x

Several special values are known:

>>> erfcinv(1)
0
>>> erfcinv(0)
oo

Differentiation with respect to x is supported:

>>> from sympy import diff
>>> diff(erfcinv(x), x)
-sqrt(pi)*exp(erfcinv(x)**2)/2

See also

erf: Gaussian error function.
erfc: Complementary error function.
erfi: Imaginary error function.
erf2: Two-argument error function.
erfinv: Inverse error function.
erf2inv: Inverse two-argument error function.

References

R287: https://en.wikipedia.org/wiki/Error_function#Inverse_functions
R288: http://functions.wolfram.com/GammaBetaErf/InverseErfc/

class sympy.functions.special.error_functions.erf2inv[source]¶

Two-argument Inverse error function. The erf2inv function is defined as:

\[\mathrm{erf2}(x, w) = y \quad \Rightarrow \quad \mathrm{erf2inv}(x, y) = w\]

Examples

>>> from sympy import I, oo, erf2inv, erfinv, erfcinv
>>> from sympy.abc import x, y

Several special values are known:

>>> erf2inv(0, 0)
0
>>> erf2inv(1, 0)
1
>>> erf2inv(0, 1)
oo
>>> erf2inv(0, y)
erfinv(y)
>>> erf2inv(oo, y)
erfcinv(-y)

Differentiation with respect to x and y is supported:

>>> from sympy import diff
>>> diff(erf2inv(x, y), x)
exp(-x**2 + erf2inv(x, y)**2)
>>> diff(erf2inv(x, y), y)
sqrt(pi)*exp(erf2inv(x, y)**2)/2

See also

erf: Gaussian error function.
erfc: Complementary error function.
erfi: Imaginary error function.
erf2: Two-argument error function.
erfinv: Inverse error function.
erfcinv: Inverse complementary error function.

References

R289: http://functions.wolfram.com/GammaBetaErf/InverseErf2/

class sympy.functions.special.error_functions.FresnelIntegral[source]¶: Base class for the Fresnel integrals.

class sympy.functions.special.error_functions.fresnels[source]¶

Fresnel integral S.

This function is defined by

\[\operatorname{S}(z) = \int_0^z \sin{\frac{\pi}{2} t^2} \mathrm{d}t.\]

It is an entire function.

Examples

>>> from sympy import I, oo, fresnels
>>> from sympy.abc import z

Several special values are known:

>>> fresnels(0)
0
>>> fresnels(oo)
1/2
>>> fresnels(-oo)
-1/2
>>> fresnels(I*oo)
-I/2
>>> fresnels(-I*oo)
I/2

In general one can pull out factors of -1 and \(i\) from the argument:

>>> fresnels(-z)
-fresnels(z)
>>> fresnels(I*z)
-I*fresnels(z)

The Fresnel S integral obeys the mirror symmetry \(\overline{S(z)} = S(\bar{z})\):

>>> from sympy import conjugate
>>> conjugate(fresnels(z))
fresnels(conjugate(z))

Differentiation with respect to \(z\) is supported:

>>> from sympy import diff
>>> diff(fresnels(z), z)
sin(pi*z**2/2)

Defining the Fresnel functions via an integral

>>> from sympy import integrate, pi, sin, gamma, expand_func
>>> integrate(sin(pi*z**2/2), z)
3*fresnels(z)*gamma(3/4)/(4*gamma(7/4))
>>> expand_func(integrate(sin(pi*z**2/2), z))
fresnels(z)

We can numerically evaluate the Fresnel integral to arbitrary precision on the whole complex plane:

>>> fresnels(2).evalf(30)
0.343415678363698242195300815958

>>> fresnels(-2*I).evalf(30)
0.343415678363698242195300815958*I

See also

fresnelc: Fresnel cosine integral.

References

R290: https://en.wikipedia.org/wiki/Fresnel_integral
R291: http://dlmf.nist.gov/7
R292: http://mathworld.wolfram.com/FresnelIntegrals.html
R293: http://functions.wolfram.com/GammaBetaErf/FresnelS
R294: The converging factors for the fresnel integrals by John W. Wrench Jr. and Vicki Alley

class sympy.functions.special.error_functions.fresnelc[source]¶

Fresnel integral C.

This function is defined by

\[\operatorname{C}(z) = \int_0^z \cos{\frac{\pi}{2} t^2} \mathrm{d}t.\]

It is an entire function.

Examples

>>> from sympy import I, oo, fresnelc
>>> from sympy.abc import z

Several special values are known:

>>> fresnelc(0)
0
>>> fresnelc(oo)
1/2
>>> fresnelc(-oo)
-1/2
>>> fresnelc(I*oo)
I/2
>>> fresnelc(-I*oo)
-I/2

In general one can pull out factors of -1 and \(i\) from the argument:

>>> fresnelc(-z)
-fresnelc(z)
>>> fresnelc(I*z)
I*fresnelc(z)

The Fresnel C integral obeys the mirror symmetry \(\overline{C(z)} = C(\bar{z})\):

>>> from sympy import conjugate
>>> conjugate(fresnelc(z))
fresnelc(conjugate(z))

Differentiation with respect to \(z\) is supported:

>>> from sympy import diff
>>> diff(fresnelc(z), z)
cos(pi*z**2/2)

Defining the Fresnel functions via an integral

>>> from sympy import integrate, pi, cos, gamma, expand_func
>>> integrate(cos(pi*z**2/2), z)
fresnelc(z)*gamma(1/4)/(4*gamma(5/4))
>>> expand_func(integrate(cos(pi*z**2/2), z))
fresnelc(z)

We can numerically evaluate the Fresnel integral to arbitrary precision on the whole complex plane:

>>> fresnelc(2).evalf(30)
0.488253406075340754500223503357

>>> fresnelc(-2*I).evalf(30)
-0.488253406075340754500223503357*I

See also

fresnels: Fresnel sine integral.

References

R295: https://en.wikipedia.org/wiki/Fresnel_integral
R296: http://dlmf.nist.gov/7
R297: http://mathworld.wolfram.com/FresnelIntegrals.html
R298: http://functions.wolfram.com/GammaBetaErf/FresnelC
R299: The converging factors for the fresnel integrals by John W. Wrench Jr. and Vicki Alley

Exponential, Logarithmic and Trigonometric Integrals¶

class sympy.functions.special.error_functions.Ei[source]¶

The classical exponential integral.

For use in SymPy, this function is defined as

\[\operatorname{Ei}(x) = \sum_{n=1}^\infty \frac{x^n}{n\, n!} + \log(x) + \gamma,\]

where \(\gamma\) is the Euler-Mascheroni constant.

If \(x\) is a polar number, this defines an analytic function on the Riemann surface of the logarithm. Otherwise this defines an analytic function in the cut plane \(\mathbb{C} \setminus (-\infty, 0]\).

Background

The name exponential integral comes from the following statement:

\[\operatorname{Ei}(x) = \int_{-\infty}^x \frac{e^t}{t} \mathrm{d}t\]

If the integral is interpreted as a Cauchy principal value, this statement holds for \(x > 0\) and \(\operatorname{Ei}(x)\) as defined above.

Examples

>>> from sympy import Ei, polar_lift, exp_polar, I, pi
>>> from sympy.abc import x

>>> Ei(-1)
Ei(-1)

This yields a real value:

>>> Ei(-1).n(chop=True)
-0.219383934395520

On the other hand the analytic continuation is not real:

>>> Ei(polar_lift(-1)).n(chop=True)
-0.21938393439552 + 3.14159265358979*I

The exponential integral has a logarithmic branch point at the origin:

>>> Ei(x*exp_polar(2*I*pi))
Ei(x) + 2*I*pi

Differentiation is supported:

>>> Ei(x).diff(x)
exp(x)/x

The exponential integral is related to many other special functions. For example:

>>> from sympy import uppergamma, expint, Shi
>>> Ei(x).rewrite(expint)
-expint(1, x*exp_polar(I*pi)) - I*pi
>>> Ei(x).rewrite(Shi)
Chi(x) + Shi(x)

See also

expint: Generalised exponential integral.
E1: Special case of the generalised exponential integral.
li: Logarithmic integral.
Li: Offset logarithmic integral.
Si: Sine integral.
Ci: Cosine integral.
Shi: Hyperbolic sine integral.
Chi: Hyperbolic cosine integral.
sympy.functions.special.gamma_functions.uppergamma: Upper incomplete gamma function.

References

R300: http://dlmf.nist.gov/6.6
R301: https://en.wikipedia.org/wiki/Exponential_integral
R302: Abramowitz & Stegun, section 5: http://people.math.sfu.ca/~cbm/aands/page_228.htm

class sympy.functions.special.error_functions.expint[source]¶

Generalized exponential integral.

This function is defined as

\[\operatorname{E}_\nu(z) = z^{\nu - 1} \Gamma(1 - \nu, z),\]

where \(\Gamma(1 - \nu, z)\) is the upper incomplete gamma function (uppergamma).

Hence for \(z\) with positive real part we have

\[\operatorname{E}_\nu(z) = \int_1^\infty \frac{e^{-zt}}{z^\nu} \mathrm{d}t,\]

which explains the name.

The representation as an incomplete gamma function provides an analytic continuation for \(\operatorname{E}_\nu(z)\). If \(\nu\) is a non-positive integer the exponential integral is thus an unbranched function of \(z\), otherwise there is a branch point at the origin. Refer to the incomplete gamma function documentation for details of the branching behavior.

Examples

>>> from sympy import expint, S
>>> from sympy.abc import nu, z

Differentiation is supported. Differentiation with respect to z explains further the name: for integral orders, the exponential integral is an iterated integral of the exponential function.

>>> expint(nu, z).diff(z)
-expint(nu - 1, z)

Differentiation with respect to nu has no classical expression:

>>> expint(nu, z).diff(nu)
-z**(nu - 1)*meijerg(((), (1, 1)), ((0, 0, 1 - nu), ()), z)

At non-postive integer orders, the exponential integral reduces to the exponential function:

>>> expint(0, z)
exp(-z)/z
>>> expint(-1, z)
exp(-z)/z + exp(-z)/z**2

At half-integers it reduces to error functions:

>>> expint(S(1)/2, z)
sqrt(pi)*erfc(sqrt(z))/sqrt(z)

At positive integer orders it can be rewritten in terms of exponentials and expint(1, z). Use expand_func() to do this:

>>> from sympy import expand_func
>>> expand_func(expint(5, z))
z**4*expint(1, z)/24 + (-z**3 + z**2 - 2*z + 6)*exp(-z)/24

The generalised exponential integral is essentially equivalent to the incomplete gamma function:

>>> from sympy import uppergamma
>>> expint(nu, z).rewrite(uppergamma)
z**(nu - 1)*uppergamma(1 - nu, z)

As such it is branched at the origin:

>>> from sympy import exp_polar, pi, I
>>> expint(4, z*exp_polar(2*pi*I))
I*pi*z**3/3 + expint(4, z)
>>> expint(nu, z*exp_polar(2*pi*I))
z**(nu - 1)*(exp(2*I*pi*nu) - 1)*gamma(1 - nu) + expint(nu, z)

See also

Ei: Another related function called exponential integral.
E1: The classical case, returns expint(1, z).
li: Logarithmic integral.
Li: Offset logarithmic integral.
Si: Sine integral.
Ci: Cosine integral.
Shi: Hyperbolic sine integral.
Chi: Hyperbolic cosine integral.

sympy.functions.special.gamma_functions.uppergamma

References

R303: http://dlmf.nist.gov/8.19
R304: http://functions.wolfram.com/GammaBetaErf/ExpIntegralE/
R305: https://en.wikipedia.org/wiki/Exponential_integral

sympy.functions.special.error_functions.E1(z)[source]¶

Classical case of the generalized exponential integral.

This is equivalent to expint(1, z).

See also

Ei: Exponential integral.
expint: Generalised exponential integral.
li: Logarithmic integral.
Li: Offset logarithmic integral.
Si: Sine integral.
Ci: Cosine integral.
Shi: Hyperbolic sine integral.
Chi: Hyperbolic cosine integral.

class sympy.functions.special.error_functions.li[source]¶

The classical logarithmic integral.

For the use in SymPy, this function is defined as

\[\operatorname{li}(x) = \int_0^x \frac{1}{\log(t)} \mathrm{d}t \,.\]

Examples

>>> from sympy import I, oo, li
>>> from sympy.abc import z

Several special values are known:

>>> li(0)
0
>>> li(1)
-oo
>>> li(oo)
oo

Differentiation with respect to z is supported:

>>> from sympy import diff
>>> diff(li(z), z)
1/log(z)

Defining the \(li\) function via an integral:

The logarithmic integral can also be defined in terms of Ei:

>>> from sympy import Ei
>>> li(z).rewrite(Ei)
Ei(log(z))
>>> diff(li(z).rewrite(Ei), z)
1/log(z)

We can numerically evaluate the logarithmic integral to arbitrary precision on the whole complex plane (except the singular points):

>>> li(2).evalf(30)
1.04516378011749278484458888919

>>> li(2*I).evalf(30)
1.0652795784357498247001125598 + 3.08346052231061726610939702133*I

We can even compute Soldner’s constant by the help of mpmath:

>>> from mpmath import findroot
>>> findroot(li, 2)
1.45136923488338

Further transformations include rewriting \(li\) in terms of the trigonometric integrals \(Si\), \(Ci\), \(Shi\) and \(Chi\):

>>> from sympy import Si, Ci, Shi, Chi
>>> li(z).rewrite(Si)
-log(I*log(z)) - log(1/log(z))/2 + log(log(z))/2 + Ci(I*log(z)) + Shi(log(z))
>>> li(z).rewrite(Ci)
-log(I*log(z)) - log(1/log(z))/2 + log(log(z))/2 + Ci(I*log(z)) + Shi(log(z))
>>> li(z).rewrite(Shi)
-log(1/log(z))/2 + log(log(z))/2 + Chi(log(z)) - Shi(log(z))
>>> li(z).rewrite(Chi)
-log(1/log(z))/2 + log(log(z))/2 + Chi(log(z)) - Shi(log(z))

See also

Li: Offset logarithmic integral.
Ei: Exponential integral.
expint: Generalised exponential integral.
E1: Special case of the generalised exponential integral.
Si: Sine integral.
Ci: Cosine integral.
Shi: Hyperbolic sine integral.
Chi: Hyperbolic cosine integral.

References

R306: https://en.wikipedia.org/wiki/Logarithmic_integral
R307: http://mathworld.wolfram.com/LogarithmicIntegral.html
R308: http://dlmf.nist.gov/6
R309: http://mathworld.wolfram.com/SoldnersConstant.html

class sympy.functions.special.error_functions.Li[source]¶

The offset logarithmic integral.

For the use in SymPy, this function is defined as

\[\operatorname{Li}(x) = \operatorname{li}(x) - \operatorname{li}(2)\]

Examples

>>> from sympy import I, oo, Li
>>> from sympy.abc import z

The following special value is known:

>>> Li(2)
0

Differentiation with respect to z is supported:

>>> from sympy import diff
>>> diff(Li(z), z)
1/log(z)

The shifted logarithmic integral can be written in terms of \(li(z)\):

>>> from sympy import li
>>> Li(z).rewrite(li)
li(z) - li(2)

We can numerically evaluate the logarithmic integral to arbitrary precision on the whole complex plane (except the singular points):

>>> Li(2).evalf(30)
0

>>> Li(4).evalf(30)
1.92242131492155809316615998938

See also

li: Logarithmic integral.
Ei: Exponential integral.
expint: Generalised exponential integral.
E1: Special case of the generalised exponential integral.
Si: Sine integral.
Ci: Cosine integral.
Shi: Hyperbolic sine integral.
Chi: Hyperbolic cosine integral.

References

R310: https://en.wikipedia.org/wiki/Logarithmic_integral
R311: http://mathworld.wolfram.com/LogarithmicIntegral.html
R312: http://dlmf.nist.gov/6

class sympy.functions.special.error_functions.Si[source]¶

Sine integral.

This function is defined by

\[\operatorname{Si}(z) = \int_0^z \frac{\sin{t}}{t} \mathrm{d}t.\]

It is an entire function.

Examples

>>> from sympy import Si
>>> from sympy.abc import z

The sine integral is an antiderivative of sin(z)/z:

>>> Si(z).diff(z)
sin(z)/z

It is unbranched:

>>> from sympy import exp_polar, I, pi
>>> Si(z*exp_polar(2*I*pi))
Si(z)

Sine integral behaves much like ordinary sine under multiplication by I:

>>> Si(I*z)
I*Shi(z)
>>> Si(-z)
-Si(z)

It can also be expressed in terms of exponential integrals, but beware that the latter is branched:

>>> from sympy import expint
>>> Si(z).rewrite(expint)
-I*(-expint(1, z*exp_polar(-I*pi/2))/2 +
     expint(1, z*exp_polar(I*pi/2))/2) + pi/2

It can be rewritten in the form of sinc function (By definition)

>>> from sympy import sinc
>>> Si(z).rewrite(sinc)
Integral(sinc(t), (t, 0, z))

See also

Ci: Cosine integral.
Shi: Hyperbolic sine integral.
Chi: Hyperbolic cosine integral.
Ei: Exponential integral.
expint: Generalised exponential integral.
sinc: unnormalized sinc function
E1: Special case of the generalised exponential integral.
li: Logarithmic integral.
Li: Offset logarithmic integral.

References

R313: https://en.wikipedia.org/wiki/Trigonometric_integral

class sympy.functions.special.error_functions.Ci[source]¶

Cosine integral.

This function is defined for positive \(x\) by

\[\operatorname{Ci}(x) = \gamma + \log{x} + \int_0^x \frac{\cos{t} - 1}{t} \mathrm{d}t = -\int_x^\infty \frac{\cos{t}}{t} \mathrm{d}t,\]

where \(\gamma\) is the Euler-Mascheroni constant.

We have

\[\operatorname{Ci}(z) = -\frac{\operatorname{E}_1\left(e^{i\pi/2} z\right) + \operatorname{E}_1\left(e^{-i \pi/2} z\right)}{2}\]

which holds for all polar \(z\) and thus provides an analytic continuation to the Riemann surface of the logarithm.

The formula also holds as stated for \(z \in \mathbb{C}\) with \(\Re(z) > 0\). By lifting to the principal branch we obtain an analytic function on the cut complex plane.

Examples

>>> from sympy import Ci
>>> from sympy.abc import z

The cosine integral is a primitive of \(\cos(z)/z\):

>>> Ci(z).diff(z)
cos(z)/z

It has a logarithmic branch point at the origin:

>>> from sympy import exp_polar, I, pi
>>> Ci(z*exp_polar(2*I*pi))
Ci(z) + 2*I*pi

The cosine integral behaves somewhat like ordinary \(\cos\) under multiplication by \(i\):

>>> from sympy import polar_lift
>>> Ci(polar_lift(I)*z)
Chi(z) + I*pi/2
>>> Ci(polar_lift(-1)*z)
Ci(z) + I*pi

It can also be expressed in terms of exponential integrals:

>>> from sympy import expint
>>> Ci(z).rewrite(expint)
-expint(1, z*exp_polar(-I*pi/2))/2 - expint(1, z*exp_polar(I*pi/2))/2

See also

Si: Sine integral.
Shi: Hyperbolic sine integral.
Chi: Hyperbolic cosine integral.
Ei: Exponential integral.
expint: Generalised exponential integral.
E1: Special case of the generalised exponential integral.
li: Logarithmic integral.
Li: Offset logarithmic integral.

References

R314: https://en.wikipedia.org/wiki/Trigonometric_integral

class sympy.functions.special.error_functions.Shi[source]¶

Sinh integral.

This function is defined by

\[\operatorname{Shi}(z) = \int_0^z \frac{\sinh{t}}{t} \mathrm{d}t.\]

It is an entire function.

Examples

>>> from sympy import Shi
>>> from sympy.abc import z

The Sinh integral is a primitive of \(\sinh(z)/z\):

>>> Shi(z).diff(z)
sinh(z)/z

It is unbranched:

>>> from sympy import exp_polar, I, pi
>>> Shi(z*exp_polar(2*I*pi))
Shi(z)

The \(\sinh\) integral behaves much like ordinary \(\sinh\) under multiplication by \(i\):

>>> Shi(I*z)
I*Si(z)
>>> Shi(-z)
-Shi(z)

It can also be expressed in terms of exponential integrals, but beware that the latter is branched:

>>> from sympy import expint
>>> Shi(z).rewrite(expint)
expint(1, z)/2 - expint(1, z*exp_polar(I*pi))/2 - I*pi/2

See also

Si: Sine integral.
Ci: Cosine integral.
Chi: Hyperbolic cosine integral.
Ei: Exponential integral.
expint: Generalised exponential integral.
E1: Special case of the generalised exponential integral.
li: Logarithmic integral.
Li: Offset logarithmic integral.

References

R315: https://en.wikipedia.org/wiki/Trigonometric_integral

class sympy.functions.special.error_functions.Chi[source]¶

Cosh integral.

This function is defined for positive \(x\) by

\[\operatorname{Chi}(x) = \gamma + \log{x} + \int_0^x \frac{\cosh{t} - 1}{t} \mathrm{d}t,\]

where \(\gamma\) is the Euler-Mascheroni constant.

We have

\[\operatorname{Chi}(z) = \operatorname{Ci}\left(e^{i \pi/2}z\right) - i\frac{\pi}{2},\]

which holds for all polar \(z\) and thus provides an analytic continuation to the Riemann surface of the logarithm. By lifting to the principal branch we obtain an analytic function on the cut complex plane.

Examples

>>> from sympy import Chi
>>> from sympy.abc import z

The \(\cosh\) integral is a primitive of \(\cosh(z)/z\):

>>> Chi(z).diff(z)
cosh(z)/z

It has a logarithmic branch point at the origin:

>>> from sympy import exp_polar, I, pi
>>> Chi(z*exp_polar(2*I*pi))
Chi(z) + 2*I*pi

The \(\cosh\) integral behaves somewhat like ordinary \(\cosh\) under multiplication by \(i\):

>>> from sympy import polar_lift
>>> Chi(polar_lift(I)*z)
Ci(z) + I*pi/2
>>> Chi(polar_lift(-1)*z)
Chi(z) + I*pi

It can also be expressed in terms of exponential integrals:

>>> from sympy import expint
>>> Chi(z).rewrite(expint)
-expint(1, z)/2 - expint(1, z*exp_polar(I*pi))/2 - I*pi/2

See also

Si: Sine integral.
Ci: Cosine integral.
Shi: Hyperbolic sine integral.
Ei: Exponential integral.
expint: Generalised exponential integral.
E1: Special case of the generalised exponential integral.
li: Logarithmic integral.
Li: Offset logarithmic integral.

References

R316: https://en.wikipedia.org/wiki/Trigonometric_integral

Bessel Type Functions¶

class sympy.functions.special.bessel.BesselBase[source]¶

Abstract base class for bessel-type functions.

This class is meant to reduce code duplication. All Bessel type functions can 1) be differentiated, and the derivatives expressed in terms of similar functions and 2) be rewritten in terms of other bessel-type functions.

Here “bessel-type functions” are assumed to have one complex parameter.

To use this base class, define class attributes _a and _b such that 2*F_n' = -_a*F_{n+1} + b*F_{n-1}.

argument¶: The argument of the bessel-type function.

order¶: The order of the bessel-type function.

class sympy.functions.special.bessel.besselj[source]¶

Bessel function of the first kind.

The Bessel \(J\) function of order \(\nu\) is defined to be the function satisfying Bessel’s differential equation

\[z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2} + z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 - \nu^2) w = 0,\]

with Laurent expansion

\[J_\nu(z) = z^\nu \left(\frac{1}{\Gamma(\nu + 1) 2^\nu} + O(z^2) \right),\]

if \(\nu\) is not a negative integer. If \(\nu=-n \in \mathbb{Z}_{<0}\) is a negative integer, then the definition is

\[J_{-n}(z) = (-1)^n J_n(z).\]

Examples

Create a Bessel function object:

>>> from sympy import besselj, jn
>>> from sympy.abc import z, n
>>> b = besselj(n, z)

Differentiate it:

>>> b.diff(z)
besselj(n - 1, z)/2 - besselj(n + 1, z)/2

Rewrite in terms of spherical Bessel functions:

>>> b.rewrite(jn)
sqrt(2)*sqrt(z)*jn(n - 1/2, z)/sqrt(pi)

Access the parameter and argument:

>>> b.order
n
>>> b.argument
z

See also

bessely, besseli, besselk

References

R317: Abramowitz, Milton; Stegun, Irene A., eds. (1965), “Chapter 9”, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
R318: Luke, Y. L. (1969), The Special Functions and Their Approximations, Volume 1
R319: https://en.wikipedia.org/wiki/Bessel_function
R320: http://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/

class sympy.functions.special.bessel.bessely[source]¶

Bessel function of the second kind.

The Bessel \(Y\) function of order \(\nu\) is defined as

\[Y_\nu(z) = \lim_{\mu \to \nu} \frac{J_\mu(z) \cos(\pi \mu) - J_{-\mu}(z)}{\sin(\pi \mu)},\]

where \(J_\mu(z)\) is the Bessel function of the first kind.

It is a solution to Bessel’s equation, and linearly independent from \(J_\nu\).

Examples

>>> from sympy import bessely, yn
>>> from sympy.abc import z, n
>>> b = bessely(n, z)
>>> b.diff(z)
bessely(n - 1, z)/2 - bessely(n + 1, z)/2
>>> b.rewrite(yn)
sqrt(2)*sqrt(z)*yn(n - 1/2, z)/sqrt(pi)

See also

besselj, besseli, besselk

References

R321: http://functions.wolfram.com/Bessel-TypeFunctions/BesselY/

class sympy.functions.special.bessel.besseli[source]¶

Modified Bessel function of the first kind.

The Bessel I function is a solution to the modified Bessel equation

\[z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2} + z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 + \nu^2)^2 w = 0.\]

It can be defined as

\[I_\nu(z) = i^{-\nu} J_\nu(iz),\]

where \(J_\nu(z)\) is the Bessel function of the first kind.

Examples

>>> from sympy import besseli
>>> from sympy.abc import z, n
>>> besseli(n, z).diff(z)
besseli(n - 1, z)/2 + besseli(n + 1, z)/2

See also

besselj, bessely, besselk

References

R322: http://functions.wolfram.com/Bessel-TypeFunctions/BesselI/

class sympy.functions.special.bessel.besselk[source]¶

Modified Bessel function of the second kind.

The Bessel K function of order \(\nu\) is defined as

\[K_\nu(z) = \lim_{\mu \to \nu} \frac{\pi}{2} \frac{I_{-\mu}(z) -I_\mu(z)}{\sin(\pi \mu)},\]

where \(I_\mu(z)\) is the modified Bessel function of the first kind.

It is a solution of the modified Bessel equation, and linearly independent from \(Y_\nu\).

Examples

>>> from sympy import besselk
>>> from sympy.abc import z, n
>>> besselk(n, z).diff(z)
-besselk(n - 1, z)/2 - besselk(n + 1, z)/2

See also

besselj, besseli, bessely

References

R323: http://functions.wolfram.com/Bessel-TypeFunctions/BesselK/

class sympy.functions.special.bessel.hankel1[source]¶

Hankel function of the first kind.

This function is defined as

\[H_\nu^{(1)} = J_\nu(z) + iY_\nu(z),\]

where \(J_\nu(z)\) is the Bessel function of the first kind, and \(Y_\nu(z)\) is the Bessel function of the second kind.

It is a solution to Bessel’s equation.

Examples

>>> from sympy import hankel1
>>> from sympy.abc import z, n
>>> hankel1(n, z).diff(z)
hankel1(n - 1, z)/2 - hankel1(n + 1, z)/2

See also

hankel2, besselj, bessely

References

R324: http://functions.wolfram.com/Bessel-TypeFunctions/HankelH1/

class sympy.functions.special.bessel.hankel2[source]¶

Hankel function of the second kind.

This function is defined as

\[H_\nu^{(2)} = J_\nu(z) - iY_\nu(z),\]

where \(J_\nu(z)\) is the Bessel function of the first kind, and \(Y_\nu(z)\) is the Bessel function of the second kind.

It is a solution to Bessel’s equation, and linearly independent from \(H_\nu^{(1)}\).

Examples

>>> from sympy import hankel2
>>> from sympy.abc import z, n
>>> hankel2(n, z).diff(z)
hankel2(n - 1, z)/2 - hankel2(n + 1, z)/2

See also

hankel1, besselj, bessely

References

R325: http://functions.wolfram.com/Bessel-TypeFunctions/HankelH2/

class sympy.functions.special.bessel.jn[source]¶

Spherical Bessel function of the first kind.

This function is a solution to the spherical Bessel equation

\[z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2} + 2z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 - \nu(\nu + 1)) w = 0.\]

It can be defined as

\[j_\nu(z) = \sqrt{\frac{\pi}{2z}} J_{\nu + \frac{1}{2}}(z),\]

where \(J_\nu(z)\) is the Bessel function of the first kind.

The spherical Bessel functions of integral order are calculated using the formula:

\[j_n(z) = f_n(z) \sin{z} + (-1)^{n+1} f_{-n-1}(z) \cos{z},\]

where the coefficients \(f_n(z)\) are available as polys.orthopolys.spherical_bessel_fn().

Examples

>>> from sympy import Symbol, jn, sin, cos, expand_func, besselj, bessely
>>> from sympy import simplify
>>> z = Symbol("z")
>>> nu = Symbol("nu", integer=True)
>>> print(expand_func(jn(0, z)))
sin(z)/z
>>> expand_func(jn(1, z)) == sin(z)/z**2 - cos(z)/z
True
>>> expand_func(jn(3, z))
(-6/z**2 + 15/z**4)*sin(z) + (1/z - 15/z**3)*cos(z)
>>> jn(nu, z).rewrite(besselj)
sqrt(2)*sqrt(pi)*sqrt(1/z)*besselj(nu + 1/2, z)/2
>>> jn(nu, z).rewrite(bessely)
(-1)**nu*sqrt(2)*sqrt(pi)*sqrt(1/z)*bessely(-nu - 1/2, z)/2
>>> jn(2, 5.2+0.3j).evalf(20)
0.099419756723640344491 - 0.054525080242173562897*I

See also

besselj, bessely, besselk, yn

References

R326: http://dlmf.nist.gov/10.47

class sympy.functions.special.bessel.yn[source]¶

Spherical Bessel function of the second kind.

This function is another solution to the spherical Bessel equation, and linearly independent from \(j_n\). It can be defined as

\[y_\nu(z) = \sqrt{\frac{\pi}{2z}} Y_{\nu + \frac{1}{2}}(z),\]

where \(Y_\nu(z)\) is the Bessel function of the second kind.

For integral orders \(n\), \(y_n\) is calculated using the formula:

\[y_n(z) = (-1)^{n+1} j_{-n-1}(z)\]

Examples

>>> from sympy import Symbol, yn, sin, cos, expand_func, besselj, bessely
>>> z = Symbol("z")
>>> nu = Symbol("nu", integer=True)
>>> print(expand_func(yn(0, z)))
-cos(z)/z
>>> expand_func(yn(1, z)) == -cos(z)/z**2-sin(z)/z
True
>>> yn(nu, z).rewrite(besselj)
(-1)**(nu + 1)*sqrt(2)*sqrt(pi)*sqrt(1/z)*besselj(-nu - 1/2, z)/2
>>> yn(nu, z).rewrite(bessely)
sqrt(2)*sqrt(pi)*sqrt(1/z)*bessely(nu + 1/2, z)/2
>>> yn(2, 5.2+0.3j).evalf(20)
0.18525034196069722536 + 0.014895573969924817587*I

See also

besselj, bessely, besselk, jn

References

R327: http://dlmf.nist.gov/10.47

sympy.functions.special.bessel.jn_zeros(n, k, method='sympy', dps=15)[source]¶

Zeros of the spherical Bessel function of the first kind.

This returns an array of zeros of jn up to the k-th zero.

method = “sympy”: uses mpmath.besseljzero()
method = “scipy”: uses the SciPy’s sph_jn and newton to find all roots, which is faster than computing the zeros using a general numerical solver, but it requires SciPy and only works with low precision floating point numbers. [The function used with method=”sympy” is a recent addition to mpmath, before that a general solver was used.]

Examples

>>> from sympy import jn_zeros
>>> jn_zeros(2, 4, dps=5)
[5.7635, 9.095, 12.323, 15.515]

Airy Functions¶

class sympy.functions.special.bessel.AiryBase[source]¶

Abstract base class for Airy functions.

This class is meant to reduce code duplication.

class sympy.functions.special.bessel.airyai[source]¶

The Airy function \(\operatorname{Ai}\) of the first kind.

The Airy function \(\operatorname{Ai}(z)\) is defined to be the function satisfying Airy’s differential equation

\[\frac{\mathrm{d}^2 w(z)}{\mathrm{d}z^2} - z w(z) = 0.\]

Equivalently, for real \(z\)

\[\operatorname{Ai}(z) := \frac{1}{\pi} \int_0^\infty \cos\left(\frac{t^3}{3} + z t\right) \mathrm{d}t.\]

Examples

Create an Airy function object:

>>> from sympy import airyai
>>> from sympy.abc import z

>>> airyai(z)
airyai(z)

Several special values are known:

>>> airyai(0)
3**(1/3)/(3*gamma(2/3))
>>> from sympy import oo
>>> airyai(oo)
0
>>> airyai(-oo)
0

The Airy function obeys the mirror symmetry:

>>> from sympy import conjugate
>>> conjugate(airyai(z))
airyai(conjugate(z))

Differentiation with respect to z is supported:

>>> from sympy import diff
>>> diff(airyai(z), z)
airyaiprime(z)
>>> diff(airyai(z), z, 2)
z*airyai(z)

Series expansion is also supported:

>>> from sympy import series
>>> series(airyai(z), z, 0, 3)
3**(5/6)*gamma(1/3)/(6*pi) - 3**(1/6)*z*gamma(2/3)/(2*pi) + O(z**3)

We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane:

>>> airyai(-2).evalf(50)
0.22740742820168557599192443603787379946077222541710

Rewrite Ai(z) in terms of hypergeometric functions:

>>> from sympy import hyper
>>> airyai(z).rewrite(hyper)
-3**(2/3)*z*hyper((), (4/3,), z**3/9)/(3*gamma(1/3)) + 3**(1/3)*hyper((), (2/3,), z**3/9)/(3*gamma(2/3))

See also

airybi: Airy function of the second kind.
airyaiprime: Derivative of the Airy function of the first kind.
airybiprime: Derivative of the Airy function of the second kind.

References

R328: https://en.wikipedia.org/wiki/Airy_function
R329: http://dlmf.nist.gov/9
R330: http://www.encyclopediaofmath.org/index.php/Airy_functions
R331: http://mathworld.wolfram.com/AiryFunctions.html

class sympy.functions.special.bessel.airybi[source]¶

The Airy function \(\operatorname{Bi}\) of the second kind.

The Airy function \(\operatorname{Bi}(z)\) is defined to be the function satisfying Airy’s differential equation

\[\frac{\mathrm{d}^2 w(z)}{\mathrm{d}z^2} - z w(z) = 0.\]

Equivalently, for real \(z\)

\[\operatorname{Bi}(z) := \frac{1}{\pi} \int_0^\infty \exp\left(-\frac{t^3}{3} + z t\right) + \sin\left(\frac{t^3}{3} + z t\right) \mathrm{d}t.\]

Examples

Create an Airy function object:

>>> from sympy import airybi
>>> from sympy.abc import z

>>> airybi(z)
airybi(z)

Several special values are known:

>>> airybi(0)
3**(5/6)/(3*gamma(2/3))
>>> from sympy import oo
>>> airybi(oo)
oo
>>> airybi(-oo)
0

The Airy function obeys the mirror symmetry:

>>> from sympy import conjugate
>>> conjugate(airybi(z))
airybi(conjugate(z))

Differentiation with respect to z is supported:

>>> from sympy import diff
>>> diff(airybi(z), z)
airybiprime(z)
>>> diff(airybi(z), z, 2)
z*airybi(z)

Series expansion is also supported:

>>> from sympy import series
>>> series(airybi(z), z, 0, 3)
3**(1/3)*gamma(1/3)/(2*pi) + 3**(2/3)*z*gamma(2/3)/(2*pi) + O(z**3)

We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane:

>>> airybi(-2).evalf(50)
-0.41230258795639848808323405461146104203453483447240

Rewrite Bi(z) in terms of hypergeometric functions:

>>> from sympy import hyper
>>> airybi(z).rewrite(hyper)
3**(1/6)*z*hyper((), (4/3,), z**3/9)/gamma(1/3) + 3**(5/6)*hyper((), (2/3,), z**3/9)/(3*gamma(2/3))

See also

airyai: Airy function of the first kind.
airyaiprime: Derivative of the Airy function of the first kind.
airybiprime: Derivative of the Airy function of the second kind.

References

R332: https://en.wikipedia.org/wiki/Airy_function
R333: http://dlmf.nist.gov/9
R334: http://www.encyclopediaofmath.org/index.php/Airy_functions
R335: http://mathworld.wolfram.com/AiryFunctions.html

class sympy.functions.special.bessel.airyaiprime[source]¶

The derivative \(\operatorname{Ai}^\prime\) of the Airy function of the first kind.

The Airy function \(\operatorname{Ai}^\prime(z)\) is defined to be the function

\[\operatorname{Ai}^\prime(z) := \frac{\mathrm{d} \operatorname{Ai}(z)}{\mathrm{d} z}.\]

Examples

Create an Airy function object:

>>> from sympy import airyaiprime
>>> from sympy.abc import z

>>> airyaiprime(z)
airyaiprime(z)

Several special values are known:

>>> airyaiprime(0)
-3**(2/3)/(3*gamma(1/3))
>>> from sympy import oo
>>> airyaiprime(oo)
0

The Airy function obeys the mirror symmetry:

>>> from sympy import conjugate
>>> conjugate(airyaiprime(z))
airyaiprime(conjugate(z))

Differentiation with respect to z is supported:

>>> from sympy import diff
>>> diff(airyaiprime(z), z)
z*airyai(z)
>>> diff(airyaiprime(z), z, 2)
z*airyaiprime(z) + airyai(z)

Series expansion is also supported:

>>> from sympy import series
>>> series(airyaiprime(z), z, 0, 3)
-3**(2/3)/(3*gamma(1/3)) + 3**(1/3)*z**2/(6*gamma(2/3)) + O(z**3)

We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane:

>>> airyaiprime(-2).evalf(50)
0.61825902074169104140626429133247528291577794512415

Rewrite Ai’(z) in terms of hypergeometric functions:

>>> from sympy import hyper
>>> airyaiprime(z).rewrite(hyper)
3**(1/3)*z**2*hyper((), (5/3,), z**3/9)/(6*gamma(2/3)) - 3**(2/3)*hyper((), (1/3,), z**3/9)/(3*gamma(1/3))

See also

airyai: Airy function of the first kind.
airybi: Airy function of the second kind.
airybiprime: Derivative of the Airy function of the second kind.

References

R336: https://en.wikipedia.org/wiki/Airy_function
R337: http://dlmf.nist.gov/9
R338: http://www.encyclopediaofmath.org/index.php/Airy_functions
R339: http://mathworld.wolfram.com/AiryFunctions.html

class sympy.functions.special.bessel.airybiprime[source]¶

The derivative \(\operatorname{Bi}^\prime\) of the Airy function of the first kind.

The Airy function \(\operatorname{Bi}^\prime(z)\) is defined to be the function

\[\operatorname{Bi}^\prime(z) := \frac{\mathrm{d} \operatorname{Bi}(z)}{\mathrm{d} z}.\]

Examples

Create an Airy function object:

>>> from sympy import airybiprime
>>> from sympy.abc import z

>>> airybiprime(z)
airybiprime(z)

Several special values are known:

>>> airybiprime(0)
3**(1/6)/gamma(1/3)
>>> from sympy import oo
>>> airybiprime(oo)
oo
>>> airybiprime(-oo)
0

The Airy function obeys the mirror symmetry:

>>> from sympy import conjugate
>>> conjugate(airybiprime(z))
airybiprime(conjugate(z))

Differentiation with respect to z is supported:

>>> from sympy import diff
>>> diff(airybiprime(z), z)
z*airybi(z)
>>> diff(airybiprime(z), z, 2)
z*airybiprime(z) + airybi(z)

Series expansion is also supported:

>>> from sympy import series
>>> series(airybiprime(z), z, 0, 3)
3**(1/6)/gamma(1/3) + 3**(5/6)*z**2/(6*gamma(2/3)) + O(z**3)

We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane:

>>> airybiprime(-2).evalf(50)
0.27879516692116952268509756941098324140300059345163

Rewrite Bi’(z) in terms of hypergeometric functions:

>>> from sympy import hyper
>>> airybiprime(z).rewrite(hyper)
3**(5/6)*z**2*hyper((), (5/3,), z**3/9)/(6*gamma(2/3)) + 3**(1/6)*hyper((), (1/3,), z**3/9)/gamma(1/3)

See also

airyai: Airy function of the first kind.
airybi: Airy function of the second kind.
airyaiprime: Derivative of the Airy function of the first kind.

References

R340: https://en.wikipedia.org/wiki/Airy_function
R341: http://dlmf.nist.gov/9
R342: http://www.encyclopediaofmath.org/index.php/Airy_functions
R343: http://mathworld.wolfram.com/AiryFunctions.html

B-Splines¶

sympy.functions.special.bsplines.bspline_basis(d, knots, n, x)[source]¶

The \(n\)-th B-spline at \(x\) of degree \(d\) with knots.

B-Splines are piecewise polynomials of degree \(d\) [R344]. They are defined on a set of knots, which is a sequence of integers or floats.

The 0th degree splines have a value of one on a single interval:

>>> from sympy import bspline_basis
>>> from sympy.abc import x
>>> d = 0
>>> knots = range(5)
>>> bspline_basis(d, knots, 0, x)
Piecewise((1, (x >= 0) & (x <= 1)), (0, True))

For a given (d, knots) there are len(knots)-d-1 B-splines defined, that are indexed by n (starting at 0).

Here is an example of a cubic B-spline:

>>> bspline_basis(3, range(5), 0, x)
Piecewise((x**3/6, (x >= 0) & (x <= 1)),
          (-x**3/2 + 2*x**2 - 2*x + 2/3,
          (x >= 1) & (x <= 2)),
          (x**3/2 - 4*x**2 + 10*x - 22/3,
          (x >= 2) & (x <= 3)),
          (-x**3/6 + 2*x**2 - 8*x + 32/3,
          (x >= 3) & (x <= 4)),
          (0, True))

By repeating knot points, you can introduce discontinuities in the B-splines and their derivatives:

>>> d = 1
>>> knots = [0, 0, 2, 3, 4]
>>> bspline_basis(d, knots, 0, x)
Piecewise((1 - x/2, (x >= 0) & (x <= 2)), (0, True))

It is quite time consuming to construct and evaluate B-splines. If you need to evaluate a B-splines many times, it is best to lambdify them first:

>>> from sympy import lambdify
>>> d = 3
>>> knots = range(10)
>>> b0 = bspline_basis(d, knots, 0, x)
>>> f = lambdify(x, b0)
>>> y = f(0.5)

See also

bsplines_basis_set

References

R344(1,2): https://en.wikipedia.org/wiki/B-spline

sympy.functions.special.bsplines.bspline_basis_set(d, knots, x)[source]¶

Return the len(knots)-d-1 B-splines at x of degree d with knots.

This function returns a list of Piecewise polynomials that are the len(knots)-d-1 B-splines of degree d for the given knots. This function calls bspline_basis(d, knots, n, x) for different values of n.

Examples

>>> from sympy import bspline_basis_set
>>> from sympy.abc import x
>>> d = 2
>>> knots = range(5)
>>> splines = bspline_basis_set(d, knots, x)
>>> splines
[Piecewise((x**2/2, (x >= 0) & (x <= 1)),
           (-x**2 + 3*x - 3/2, (x >= 1) & (x <= 2)),
           (x**2/2 - 3*x + 9/2, (x >= 2) & (x <= 3)),
           (0, True)),
Piecewise((x**2/2 - x + 1/2, (x >= 1) & (x <= 2)),
          (-x**2 + 5*x - 11/2, (x >= 2) & (x <= 3)),
          (x**2/2 - 4*x + 8, (x >= 3) & (x <= 4)),
          (0, True))]

See also

bsplines_basis

Riemann Zeta and Related Functions¶

class sympy.functions.special.zeta_functions.zeta[source]¶

Hurwitz zeta function (or Riemann zeta function).

For \(\operatorname{Re}(a) > 0\) and \(\operatorname{Re}(s) > 1\), this function is defined as

\[\zeta(s, a) = \sum_{n=0}^\infty \frac{1}{(n + a)^s},\]

where the standard choice of argument for \(n + a\) is used. For fixed \(a\) with \(\operatorname{Re}(a) > 0\) the Hurwitz zeta function admits a meromorphic continuation to all of \(\mathbb{C}\), it is an unbranched function with a simple pole at \(s = 1\).

Analytic continuation to other \(a\) is possible under some circumstances, but this is not typically done.

The Hurwitz zeta function is a special case of the Lerch transcendent:

\[\zeta(s, a) = \Phi(1, s, a).\]

This formula defines an analytic continuation for all possible values of \(s\) and \(a\) (also \(\operatorname{Re}(a) < 0\)), see the documentation of lerchphi for a description of the branching behavior.

If no value is passed for \(a\), by this function assumes a default value of \(a = 1\), yielding the Riemann zeta function.

Examples

For \(a = 1\) the Hurwitz zeta function reduces to the famous Riemann zeta function:

\[\zeta(s, 1) = \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}.\]

>>> from sympy import zeta
>>> from sympy.abc import s
>>> zeta(s, 1)
zeta(s)
>>> zeta(s)
zeta(s)

The Riemann zeta function can also be expressed using the Dirichlet eta function:

>>> from sympy import dirichlet_eta
>>> zeta(s).rewrite(dirichlet_eta)
dirichlet_eta(s)/(1 - 2**(1 - s))

The Riemann zeta function at positive even integer and negative odd integer values is related to the Bernoulli numbers:

>>> zeta(2)
pi**2/6
>>> zeta(4)
pi**4/90
>>> zeta(-1)
-1/12

The specific formulae are:

\[\zeta(2n) = (-1)^{n+1} \frac{B_{2n} (2\pi)^{2n}}{2(2n)!}\]

\[\zeta(-n) = -\frac{B_{n+1}}{n+1}\]

At negative even integers the Riemann zeta function is zero:

>>> zeta(-4)
0

No closed-form expressions are known at positive odd integers, but numerical evaluation is possible:

>>> zeta(3).n()
1.20205690315959

The derivative of \(\zeta(s, a)\) with respect to \(a\) is easily computed:

>>> from sympy.abc import a
>>> zeta(s, a).diff(a)
-s*zeta(s + 1, a)

However the derivative with respect to \(s\) has no useful closed form expression:

>>> zeta(s, a).diff(s)
Derivative(zeta(s, a), s)

The Hurwitz zeta function can be expressed in terms of the Lerch transcendent, sympy.functions.special.lerchphi:

>>> from sympy import lerchphi
>>> zeta(s, a).rewrite(lerchphi)
lerchphi(1, s, a)

See also

dirichlet_eta, lerchphi, polylog

References

R345: http://dlmf.nist.gov/25.11
R346: https://en.wikipedia.org/wiki/Hurwitz_zeta_function

class sympy.functions.special.zeta_functions.dirichlet_eta[source]¶

Dirichlet eta function.

For \(\operatorname{Re}(s) > 0\), this function is defined as

\[\eta(s) = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^s}.\]

It admits a unique analytic continuation to all of \(\mathbb{C}\). It is an entire, unbranched function.

Examples

The Dirichlet eta function is closely related to the Riemann zeta function:

>>> from sympy import dirichlet_eta, zeta
>>> from sympy.abc import s
>>> dirichlet_eta(s).rewrite(zeta)
(1 - 2**(1 - s))*zeta(s)

See also

zeta

References

R347: https://en.wikipedia.org/wiki/Dirichlet_eta_function

class sympy.functions.special.zeta_functions.polylog[source]¶

Polylogarithm function.

For \(|z| < 1\) and \(s \in \mathbb{C}\), the polylogarithm is defined by

\[\operatorname{Li}_s(z) = \sum_{n=1}^\infty \frac{z^n}{n^s},\]

where the standard branch of the argument is used for \(n\). It admits an analytic continuation which is branched at \(z=1\) (notably not on the sheet of initial definition), \(z=0\) and \(z=\infty\).

The name polylogarithm comes from the fact that for \(s=1\), the polylogarithm is related to the ordinary logarithm (see examples), and that

\[\operatorname{Li}_{s+1}(z) = \int_0^z \frac{\operatorname{Li}_s(t)}{t} \mathrm{d}t.\]

The polylogarithm is a special case of the Lerch transcendent:

\[\operatorname{Li}_{s}(z) = z \Phi(z, s, 1)\]

Examples

For \(z \in \{0, 1, -1\}\), the polylogarithm is automatically expressed using other functions:

>>> from sympy import polylog
>>> from sympy.abc import s
>>> polylog(s, 0)
0
>>> polylog(s, 1)
zeta(s)
>>> polylog(s, -1)
-dirichlet_eta(s)

If \(s\) is a negative integer, \(0\) or \(1\), the polylogarithm can be expressed using elementary functions. This can be done using expand_func():

>>> from sympy import expand_func
>>> from sympy.abc import z
>>> expand_func(polylog(1, z))
-log(1 - z)
>>> expand_func(polylog(0, z))
z/(1 - z)

The derivative with respect to \(z\) can be computed in closed form:

>>> polylog(s, z).diff(z)
polylog(s - 1, z)/z

The polylogarithm can be expressed in terms of the lerch transcendent:

>>> from sympy import lerchphi
>>> polylog(s, z).rewrite(lerchphi)
z*lerchphi(z, s, 1)

See also

zeta, lerchphi

class sympy.functions.special.zeta_functions.lerchphi[source]¶

Lerch transcendent (Lerch phi function).

For \(\operatorname{Re}(a) > 0\), \(|z| < 1\) and \(s \in \mathbb{C}\), the Lerch transcendent is defined as

\[\Phi(z, s, a) = \sum_{n=0}^\infty \frac{z^n}{(n + a)^s},\]

where the standard branch of the argument is used for \(n + a\), and by analytic continuation for other values of the parameters.

A commonly used related function is the Lerch zeta function, defined by

\[L(q, s, a) = \Phi(e^{2\pi i q}, s, a).\]

Analytic Continuation and Branching Behavior

It can be shown that

\[\Phi(z, s, a) = z\Phi(z, s, a+1) + a^{-s}.\]

This provides the analytic continuation to \(\operatorname{Re}(a) \le 0\).

Assume now \(\operatorname{Re}(a) > 0\). The integral representation

\[\Phi_0(z, s, a) = \int_0^\infty \frac{t^{s-1} e^{-at}}{1 - ze^{-t}} \frac{\mathrm{d}t}{\Gamma(s)}\]

provides an analytic continuation to \(\mathbb{C} - [1, \infty)\). Finally, for \(x \in (1, \infty)\) we find

\[\lim_{\epsilon \to 0^+} \Phi_0(x + i\epsilon, s, a) -\lim_{\epsilon \to 0^+} \Phi_0(x - i\epsilon, s, a) = \frac{2\pi i \log^{s-1}{x}}{x^a \Gamma(s)},\]

using the standard branch for both \(\log{x}\) and \(\log{\log{x}}\) (a branch of \(\log{\log{x}}\) is needed to evaluate \(\log{x}^{s-1}\)). This concludes the analytic continuation. The Lerch transcendent is thus branched at \(z \in \{0, 1, \infty\}\) and \(a \in \mathbb{Z}_{\le 0}\). For fixed \(z, a\) outside these branch points, it is an entire function of \(s\).

Examples

The Lerch transcendent is a fairly general function, for this reason it does not automatically evaluate to simpler functions. Use expand_func() to achieve this.

If \(z=1\), the Lerch transcendent reduces to the Hurwitz zeta function:

>>> from sympy import lerchphi, expand_func
>>> from sympy.abc import z, s, a
>>> expand_func(lerchphi(1, s, a))
zeta(s, a)

More generally, if \(z\) is a root of unity, the Lerch transcendent reduces to a sum of Hurwitz zeta functions:

>>> expand_func(lerchphi(-1, s, a))
2**(-s)*zeta(s, a/2) - 2**(-s)*zeta(s, a/2 + 1/2)

If \(a=1\), the Lerch transcendent reduces to the polylogarithm:

>>> expand_func(lerchphi(z, s, 1))
polylog(s, z)/z

More generally, if \(a\) is rational, the Lerch transcendent reduces to a sum of polylogarithms:

>>> from sympy import S
>>> expand_func(lerchphi(z, s, S(1)/2))
2**(s - 1)*(polylog(s, sqrt(z))/sqrt(z) -
            polylog(s, sqrt(z)*exp_polar(I*pi))/sqrt(z))
>>> expand_func(lerchphi(z, s, S(3)/2))
-2**s/z + 2**(s - 1)*(polylog(s, sqrt(z))/sqrt(z) -
                      polylog(s, sqrt(z)*exp_polar(I*pi))/sqrt(z))/z

The derivatives with respect to \(z\) and \(a\) can be computed in closed form:

>>> lerchphi(z, s, a).diff(z)
(-a*lerchphi(z, s, a) + lerchphi(z, s - 1, a))/z
>>> lerchphi(z, s, a).diff(a)
-s*lerchphi(z, s + 1, a)

See also

polylog, zeta

References

R348: Bateman, H.; Erdelyi, A. (1953), Higher Transcendental Functions, Vol. I, New York: McGraw-Hill. Section 1.11.
R349: http://dlmf.nist.gov/25.14
R350: https://en.wikipedia.org/wiki/Lerch_transcendent

class sympy.functions.special.zeta_functions.stieltjes[source]¶

Represents Stieltjes constants, \(\gamma_{k}\) that occur in Laurent Series expansion of the Riemann zeta function.

Examples

>>> from sympy import stieltjes
>>> from sympy.abc import n, m
>>> stieltjes(n)
stieltjes(n)

zero’th stieltjes constant

>>> stieltjes(0)
EulerGamma
>>> stieltjes(0, 1)
EulerGamma

For generalized stieltjes constants

>>> stieltjes(n, m)
stieltjes(n, m)

Constants are only defined for integers >= 0

>>> stieltjes(-1)
zoo

References

R351: https://en.wikipedia.org/wiki/Stieltjes_constants

Hypergeometric Functions¶

class sympy.functions.special.hyper.hyper[source]¶

The (generalized) hypergeometric function is defined by a series where the ratios of successive terms are a rational function of the summation index. When convergent, it is continued analytically to the largest possible domain.

The hypergeometric function depends on two vectors of parameters, called the numerator parameters \(a_p\), and the denominator parameters \(b_q\). It also has an argument \(z\). The series definition is

\[\begin{split}{}_pF_q\left(\begin{matrix} a_1, \cdots, a_p \\ b_1, \cdots, b_q \end{matrix} \middle| z \right) = \sum_{n=0}^\infty \frac{(a_1)_n \cdots (a_p)_n}{(b_1)_n \cdots (b_q)_n} \frac{z^n}{n!},\end{split}\]

where \((a)_n = (a)(a+1)\cdots(a+n-1)\) denotes the rising factorial.

If one of the \(b_q\) is a non-positive integer then the series is undefined unless one of the \(a_p\) is a larger (i.e. smaller in magnitude) non-positive integer. If none of the \(b_q\) is a non-positive integer and one of the \(a_p\) is a non-positive integer, then the series reduces to a polynomial. To simplify the following discussion, we assume that none of the \(a_p\) or \(b_q\) is a non-positive integer. For more details, see the references.

The series converges for all \(z\) if \(p \le q\), and thus defines an entire single-valued function in this case. If \(p = q+1\) the series converges for \(|z| < 1\), and can be continued analytically into a half-plane. If \(p > q+1\) the series is divergent for all \(z\).

Note: The hypergeometric function constructor currently does not check if the parameters actually yield a well-defined function.

Examples

The parameters \(a_p\) and \(b_q\) can be passed as arbitrary iterables, for example:

>>> from sympy.functions import hyper
>>> from sympy.abc import x, n, a
>>> hyper((1, 2, 3), [3, 4], x)
hyper((1, 2, 3), (3, 4), x)

There is also pretty printing (it looks better using unicode):

>>> from sympy import pprint
>>> pprint(hyper((1, 2, 3), [3, 4], x), use_unicode=False)
  _
 |_  /1, 2, 3 |  \
 |   |        | x|
3  2 \  3, 4  |  /

The parameters must always be iterables, even if they are vectors of length one or zero:

>>> hyper((1, ), [], x)
hyper((1,), (), x)

But of course they may be variables (but if they depend on x then you should not expect much implemented functionality):

>>> hyper((n, a), (n**2,), x)
hyper((n, a), (n**2,), x)

The hypergeometric function generalizes many named special functions. The function hyperexpand() tries to express a hypergeometric function using named special functions. For example:

>>> from sympy import hyperexpand
>>> hyperexpand(hyper([], [], x))
exp(x)

You can also use expand_func:

>>> from sympy import expand_func
>>> expand_func(x*hyper([1, 1], [2], -x))
log(x + 1)

More examples:

>>> from sympy import S
>>> hyperexpand(hyper([], [S(1)/2], -x**2/4))
cos(x)
>>> hyperexpand(x*hyper([S(1)/2, S(1)/2], [S(3)/2], x**2))
asin(x)

We can also sometimes hyperexpand parametric functions:

>>> from sympy.abc import a
>>> hyperexpand(hyper([-a], [], x))
(1 - x)**a

References

R352: Luke, Y. L. (1969), The Special Functions and Their Approximations, Volume 1
R353: https://en.wikipedia.org/wiki/Generalized_hypergeometric_function

ap¶: Numerator parameters of the hypergeometric function.

argument¶: Argument of the hypergeometric function.

bq¶: Denominator parameters of the hypergeometric function.

convergence_statement¶: Return a condition on z under which the series converges.

eta¶: A quantity related to the convergence of the series.

radius_of_convergence¶

Compute the radius of convergence of the defining series.

Note that even if this is not oo, the function may still be evaluated outside of the radius of convergence by analytic continuation. But if this is zero, then the function is not actually defined anywhere else.

>>> from sympy.functions import hyper
>>> from sympy.abc import z
>>> hyper((1, 2), [3], z).radius_of_convergence
1
>>> hyper((1, 2, 3), [4], z).radius_of_convergence
0
>>> hyper((1, 2), (3, 4), z).radius_of_convergence
oo

class sympy.functions.special.hyper.meijerg[source]¶

The Meijer G-function is defined by a Mellin-Barnes type integral that resembles an inverse Mellin transform. It generalizes the hypergeometric functions.

The Meijer G-function depends on four sets of parameters. There are “numerator parameters” \(a_1, \ldots, a_n\) and \(a_{n+1}, \ldots, a_p\), and there are “denominator parameters” \(b_1, \ldots, b_m\) and \(b_{m+1}, \ldots, b_q\). Confusingly, it is traditionally denoted as follows (note the position of \(m\), \(n\), \(p\), \(q\), and how they relate to the lengths of the four parameter vectors):

\[\begin{split}G_{p,q}^{m,n} \left(\begin{matrix}a_1, \cdots, a_n & a_{n+1}, \cdots, a_p \\ b_1, \cdots, b_m & b_{m+1}, \cdots, b_q \end{matrix} \middle| z \right).\end{split}\]

However, in sympy the four parameter vectors are always available separately (see examples), so that there is no need to keep track of the decorating sub- and super-scripts on the G symbol.

The G function is defined as the following integral:

\[\frac{1}{2 \pi i} \int_L \frac{\prod_{j=1}^m \Gamma(b_j - s) \prod_{j=1}^n \Gamma(1 - a_j + s)}{\prod_{j=m+1}^q \Gamma(1- b_j +s) \prod_{j=n+1}^p \Gamma(a_j - s)} z^s \mathrm{d}s,\]

where \(\Gamma(z)\) is the gamma function. There are three possible contours which we will not describe in detail here (see the references). If the integral converges along more than one of them the definitions agree. The contours all separate the poles of \(\Gamma(1-a_j+s)\) from the poles of \(\Gamma(b_k-s)\), so in particular the G function is undefined if \(a_j - b_k \in \mathbb{Z}_{>0}\) for some \(j \le n\) and \(k \le m\).

The conditions under which one of the contours yields a convergent integral are complicated and we do not state them here, see the references.

Note: Currently the Meijer G-function constructor does not check any convergence conditions.

Examples

You can pass the parameters either as four separate vectors:

>>> from sympy.functions import meijerg
>>> from sympy.abc import x, a
>>> from sympy.core.containers import Tuple
>>> from sympy import pprint
>>> pprint(meijerg((1, 2), (a, 4), (5,), [], x), use_unicode=False)
 __1, 2 /1, 2  a, 4 |  \
/__     |           | x|
\_|4, 1 \ 5         |  /

or as two nested vectors:

>>> pprint(meijerg([(1, 2), (3, 4)], ([5], Tuple()), x), use_unicode=False)
 __1, 2 /1, 2  3, 4 |  \
/__     |           | x|
\_|4, 1 \ 5         |  /

As with the hypergeometric function, the parameters may be passed as arbitrary iterables. Vectors of length zero and one also have to be passed as iterables. The parameters need not be constants, but if they depend on the argument then not much implemented functionality should be expected.

All the subvectors of parameters are available:

>>> from sympy import pprint
>>> g = meijerg([1], [2], [3], [4], x)
>>> pprint(g, use_unicode=False)
 __1, 1 /1  2 |  \
/__     |     | x|
\_|2, 2 \3  4 |  /
>>> g.an
(1,)
>>> g.ap
(1, 2)
>>> g.aother
(2,)
>>> g.bm
(3,)
>>> g.bq
(3, 4)
>>> g.bother
(4,)

The Meijer G-function generalizes the hypergeometric functions. In some cases it can be expressed in terms of hypergeometric functions, using Slater’s theorem. For example:

>>> from sympy import hyperexpand
>>> from sympy.abc import a, b, c
>>> hyperexpand(meijerg([a], [], [c], [b], x), allow_hyper=True)
x**c*gamma(-a + c + 1)*hyper((-a + c + 1,),
                             (-b + c + 1,), -x)/gamma(-b + c + 1)

Thus the Meijer G-function also subsumes many named functions as special cases. You can use expand_func or hyperexpand to (try to) rewrite a Meijer G-function in terms of named special functions. For example:

>>> from sympy import expand_func, S
>>> expand_func(meijerg([[],[]], [[0],[]], -x))
exp(x)
>>> hyperexpand(meijerg([[],[]], [[S(1)/2],[0]], (x/2)**2))
sin(x)/sqrt(pi)

See also

hyper, sympy.simplify.hyperexpand

References

R354: Luke, Y. L. (1969), The Special Functions and Their Approximations, Volume 1
R355: https://en.wikipedia.org/wiki/Meijer_G-function

an¶: First set of numerator parameters.

aother¶: Second set of numerator parameters.

ap¶: Combined numerator parameters.

argument¶: Argument of the Meijer G-function.

bm¶: First set of denominator parameters.

bother¶: Second set of denominator parameters.

bq¶: Combined denominator parameters.

delta¶: A quantity related to the convergence region of the integral, c.f. references.

get_period()[source]¶

Return a number P such that G(x*exp(I*P)) == G(x).

>>> from sympy.functions.special.hyper import meijerg
>>> from sympy.abc import z
>>> from sympy import pi, S

>>> meijerg([1], [], [], [], z).get_period()
2*pi
>>> meijerg([pi], [], [], [], z).get_period()
oo
>>> meijerg([1, 2], [], [], [], z).get_period()
oo
>>> meijerg([1,1], [2], [1, S(1)/2, S(1)/3], [1], z).get_period()
12*pi

integrand(s)[source]¶: Get the defining integrand D(s).

is_number¶: Returns true if expression has numeric data only.

nu¶: A quantity related to the convergence region of the integral, c.f. references.

Elliptic integrals¶

class sympy.functions.special.elliptic_integrals.elliptic_k[source]¶

The complete elliptic integral of the first kind, defined by

\[K(m) = F\left(\tfrac{\pi}{2}\middle| m\right)\]

where \(F\left(z\middle| m\right)\) is the Legendre incomplete elliptic integral of the first kind.

The function \(K(m)\) is a single-valued function on the complex plane with branch cut along the interval \((1, \infty)\).

Note that our notation defines the incomplete elliptic integral in terms of the parameter \(m\) instead of the elliptic modulus (eccentricity) \(k\). In this case, the parameter \(m\) is defined as \(m=k^2\).

Examples

>>> from sympy import elliptic_k, I, pi
>>> from sympy.abc import m
>>> elliptic_k(0)
pi/2
>>> elliptic_k(1.0 + I)
1.50923695405127 + 0.625146415202697*I
>>> elliptic_k(m).series(n=3)
pi/2 + pi*m/8 + 9*pi*m**2/128 + O(m**3)

See also

elliptic_f

References

R356: https://en.wikipedia.org/wiki/Elliptic_integrals
R357: http://functions.wolfram.com/EllipticIntegrals/EllipticK

class sympy.functions.special.elliptic_integrals.elliptic_f[source]¶

The Legendre incomplete elliptic integral of the first kind, defined by

\[F\left(z\middle| m\right) = \int_0^z \frac{dt}{\sqrt{1 - m \sin^2 t}}\]

This function reduces to a complete elliptic integral of the first kind, \(K(m)\), when \(z = \pi/2\).

Note that our notation defines the incomplete elliptic integral in terms of the parameter \(m\) instead of the elliptic modulus (eccentricity) \(k\). In this case, the parameter \(m\) is defined as \(m=k^2\).

Examples

>>> from sympy import elliptic_f, I, O
>>> from sympy.abc import z, m
>>> elliptic_f(z, m).series(z)
z + z**5*(3*m**2/40 - m/30) + m*z**3/6 + O(z**6)
>>> elliptic_f(3.0 + I/2, 1.0 + I)
2.909449841483 + 1.74720545502474*I

See also

elliptic_k

References

R358: https://en.wikipedia.org/wiki/Elliptic_integrals
R359: http://functions.wolfram.com/EllipticIntegrals/EllipticF

class sympy.functions.special.elliptic_integrals.elliptic_e[source]¶

Called with two arguments \(z\) and \(m\), evaluates the incomplete elliptic integral of the second kind, defined by

\[E\left(z\middle| m\right) = \int_0^z \sqrt{1 - m \sin^2 t} dt\]

Called with a single argument \(m\), evaluates the Legendre complete elliptic integral of the second kind

\[E(m) = E\left(\tfrac{\pi}{2}\middle| m\right)\]

The function \(E(m)\) is a single-valued function on the complex plane with branch cut along the interval \((1, \infty)\).

Note that our notation defines the incomplete elliptic integral in terms of the parameter \(m\) instead of the elliptic modulus (eccentricity) \(k\). In this case, the parameter \(m\) is defined as \(m=k^2\).

Examples

>>> from sympy import elliptic_e, I, pi, O
>>> from sympy.abc import z, m
>>> elliptic_e(z, m).series(z)
z + z**5*(-m**2/40 + m/30) - m*z**3/6 + O(z**6)
>>> elliptic_e(m).series(n=4)
pi/2 - pi*m/8 - 3*pi*m**2/128 - 5*pi*m**3/512 + O(m**4)
>>> elliptic_e(1 + I, 2 - I/2).n()
1.55203744279187 + 0.290764986058437*I
>>> elliptic_e(0)
pi/2
>>> elliptic_e(2.0 - I)
0.991052601328069 + 0.81879421395609*I

References

R360: https://en.wikipedia.org/wiki/Elliptic_integrals
R361: http://functions.wolfram.com/EllipticIntegrals/EllipticE2
R362: http://functions.wolfram.com/EllipticIntegrals/EllipticE

class sympy.functions.special.elliptic_integrals.elliptic_pi[source]¶

Called with three arguments \(n\), \(z\) and \(m\), evaluates the Legendre incomplete elliptic integral of the third kind, defined by

\[\Pi\left(n; z\middle| m\right) = \int_0^z \frac{dt} {\left(1 - n \sin^2 t\right) \sqrt{1 - m \sin^2 t}}\]

Called with two arguments \(n\) and \(m\), evaluates the complete elliptic integral of the third kind:

\[\Pi\left(n\middle| m\right) = \Pi\left(n; \tfrac{\pi}{2}\middle| m\right)\]

Note that our notation defines the incomplete elliptic integral in terms of the parameter \(m\) instead of the elliptic modulus (eccentricity) \(k\). In this case, the parameter \(m\) is defined as \(m=k^2\).

Examples

>>> from sympy import elliptic_pi, I, pi, O, S
>>> from sympy.abc import z, n, m
>>> elliptic_pi(n, z, m).series(z, n=4)
z + z**3*(m/6 + n/3) + O(z**4)
>>> elliptic_pi(0.5 + I, 1.0 - I, 1.2)
2.50232379629182 - 0.760939574180767*I
>>> elliptic_pi(0, 0)
pi/2
>>> elliptic_pi(1.0 - I/3, 2.0 + I)
3.29136443417283 + 0.32555634906645*I

References

R363: https://en.wikipedia.org/wiki/Elliptic_integrals
R364: http://functions.wolfram.com/EllipticIntegrals/EllipticPi3
R365: http://functions.wolfram.com/EllipticIntegrals/EllipticPi

Mathieu Functions¶

class sympy.functions.special.mathieu_functions.MathieuBase[source]¶

Abstract base class for Mathieu functions.

This class is meant to reduce code duplication.

class sympy.functions.special.mathieu_functions.mathieus[source]¶

The Mathieu Sine function \(S(a,q,z)\). This function is one solution of the Mathieu differential equation:

\[y(x)^{\prime\prime} + (a - 2 q \cos(2 x)) y(x) = 0\]

The other solution is the Mathieu Cosine function.

Examples

>>> from sympy import diff, mathieus
>>> from sympy.abc import a, q, z

>>> mathieus(a, q, z)
mathieus(a, q, z)

>>> mathieus(a, 0, z)
sin(sqrt(a)*z)

>>> diff(mathieus(a, q, z), z)
mathieusprime(a, q, z)

See also

mathieuc: Mathieu cosine function.
mathieusprime: Derivative of Mathieu sine function.
mathieucprime: Derivative of Mathieu cosine function.

References

R366: https://en.wikipedia.org/wiki/Mathieu_function
R367: http://dlmf.nist.gov/28
R368: http://mathworld.wolfram.com/MathieuBase.html
R369: http://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuS/

class sympy.functions.special.mathieu_functions.mathieuc[source]¶

The Mathieu Cosine function \(C(a,q,z)\). This function is one solution of the Mathieu differential equation:

\[y(x)^{\prime\prime} + (a - 2 q \cos(2 x)) y(x) = 0\]

The other solution is the Mathieu Sine function.

Examples

>>> from sympy import diff, mathieuc
>>> from sympy.abc import a, q, z

>>> mathieuc(a, q, z)
mathieuc(a, q, z)

>>> mathieuc(a, 0, z)
cos(sqrt(a)*z)

>>> diff(mathieuc(a, q, z), z)
mathieucprime(a, q, z)

See also

mathieus: Mathieu sine function
mathieusprime: Derivative of Mathieu sine function
mathieucprime: Derivative of Mathieu cosine function

References

R370: https://en.wikipedia.org/wiki/Mathieu_function
R371: http://dlmf.nist.gov/28
R372: http://mathworld.wolfram.com/MathieuBase.html
R373: http://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuC/

class sympy.functions.special.mathieu_functions.mathieusprime[source]¶

The derivative \(S^{\prime}(a,q,z)\) of the Mathieu Sine function. This function is one solution of the Mathieu differential equation:

\[y(x)^{\prime\prime} + (a - 2 q \cos(2 x)) y(x) = 0\]

The other solution is the Mathieu Cosine function.

Examples

>>> from sympy import diff, mathieusprime
>>> from sympy.abc import a, q, z

>>> mathieusprime(a, q, z)
mathieusprime(a, q, z)

>>> mathieusprime(a, 0, z)
sqrt(a)*cos(sqrt(a)*z)

>>> diff(mathieusprime(a, q, z), z)
(-a + 2*q*cos(2*z))*mathieus(a, q, z)

See also

mathieus: Mathieu sine function
mathieuc: Mathieu cosine function
mathieucprime: Derivative of Mathieu cosine function

References

R374: https://en.wikipedia.org/wiki/Mathieu_function
R375: http://dlmf.nist.gov/28
R376: http://mathworld.wolfram.com/MathieuBase.html
R377: http://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuSPrime/

class sympy.functions.special.mathieu_functions.mathieucprime[source]¶

The derivative \(C^{\prime}(a,q,z)\) of the Mathieu Cosine function. This function is one solution of the Mathieu differential equation:

\[y(x)^{\prime\prime} + (a - 2 q \cos(2 x)) y(x) = 0\]

The other solution is the Mathieu Sine function.

Examples

>>> from sympy import diff, mathieucprime
>>> from sympy.abc import a, q, z

>>> mathieucprime(a, q, z)
mathieucprime(a, q, z)

>>> mathieucprime(a, 0, z)
-sqrt(a)*sin(sqrt(a)*z)

>>> diff(mathieucprime(a, q, z), z)
(-a + 2*q*cos(2*z))*mathieuc(a, q, z)

See also

mathieus: Mathieu sine function
mathieuc: Mathieu cosine function
mathieusprime: Derivative of Mathieu sine function

References

R378: https://en.wikipedia.org/wiki/Mathieu_function
R379: http://dlmf.nist.gov/28
R380: http://mathworld.wolfram.com/MathieuBase.html
R381: http://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuCPrime/

Orthogonal Polynomials¶

This module mainly implements special orthogonal polynomials.

See also functions.combinatorial.numbers which contains some combinatorial polynomials.

Jacobi Polynomials¶

class sympy.functions.special.polynomials.jacobi[source]¶

Jacobi polynomial \(P_n^{\left(\alpha, \beta\right)}(x)\)

jacobi(n, alpha, beta, x) gives the nth Jacobi polynomial in x, \(P_n^{\left(\alpha, \beta\right)}(x)\).

The Jacobi polynomials are orthogonal on \([-1, 1]\) with respect to the weight \(\left(1-x\right)^\alpha \left(1+x\right)^\beta\).

Examples

>>> from sympy import jacobi, S, conjugate, diff
>>> from sympy.abc import n,a,b,x

>>> jacobi(0, a, b, x)
1
>>> jacobi(1, a, b, x)
a/2 - b/2 + x*(a/2 + b/2 + 1)
>>> jacobi(2, a, b, x)   # doctest:+SKIP
(a**2/8 - a*b/4 - a/8 + b**2/8 - b/8 + x**2*(a**2/8 + a*b/4 + 7*a/8 +
b**2/8 + 7*b/8 + 3/2) + x*(a**2/4 + 3*a/4 - b**2/4 - 3*b/4) - 1/2)

>>> jacobi(n, a, b, x)
jacobi(n, a, b, x)

>>> jacobi(n, a, a, x)
RisingFactorial(a + 1, n)*gegenbauer(n,
    a + 1/2, x)/RisingFactorial(2*a + 1, n)

>>> jacobi(n, 0, 0, x)
legendre(n, x)

>>> jacobi(n, S(1)/2, S(1)/2, x)
RisingFactorial(3/2, n)*chebyshevu(n, x)/factorial(n + 1)

>>> jacobi(n, -S(1)/2, -S(1)/2, x)
RisingFactorial(1/2, n)*chebyshevt(n, x)/factorial(n)

>>> jacobi(n, a, b, -x)
(-1)**n*jacobi(n, b, a, x)

>>> jacobi(n, a, b, 0)
2**(-n)*gamma(a + n + 1)*hyper((-b - n, -n), (a + 1,), -1)/(factorial(n)*gamma(a + 1))
>>> jacobi(n, a, b, 1)
RisingFactorial(a + 1, n)/factorial(n)

>>> conjugate(jacobi(n, a, b, x))
jacobi(n, conjugate(a), conjugate(b), conjugate(x))

>>> diff(jacobi(n,a,b,x), x)
(a/2 + b/2 + n/2 + 1/2)*jacobi(n - 1, a + 1, b + 1, x)

References

R382: https://en.wikipedia.org/wiki/Jacobi_polynomials
R383: http://mathworld.wolfram.com/JacobiPolynomial.html
R384: http://functions.wolfram.com/Polynomials/JacobiP/

sympy.functions.special.polynomials.jacobi_normalized(n, a, b, x)[source]¶

Jacobi polynomial \(P_n^{\left(\alpha, \beta\right)}(x)\)

jacobi_normalized(n, alpha, beta, x) gives the nth Jacobi polynomial in x, \(P_n^{\left(\alpha, \beta\right)}(x)\).

The Jacobi polynomials are orthogonal on \([-1, 1]\) with respect to the weight \(\left(1-x\right)^\alpha \left(1+x\right)^\beta\).

This functions returns the polynomials normilzed:

\[\int_{-1}^{1} P_m^{\left(\alpha, \beta\right)}(x) P_n^{\left(\alpha, \beta\right)}(x) (1-x)^{\alpha} (1+x)^{\beta} \mathrm{d}x = \delta_{m,n}\]

Examples

>>> from sympy import jacobi_normalized
>>> from sympy.abc import n,a,b,x

>>> jacobi_normalized(n, a, b, x)
jacobi(n, a, b, x)/sqrt(2**(a + b + 1)*gamma(a + n + 1)*gamma(b + n + 1)/((a + b + 2*n + 1)*factorial(n)*gamma(a + b + n + 1)))

References

R385: https://en.wikipedia.org/wiki/Jacobi_polynomials
R386: http://mathworld.wolfram.com/JacobiPolynomial.html
R387: http://functions.wolfram.com/Polynomials/JacobiP/

Gegenbauer Polynomials¶

class sympy.functions.special.polynomials.gegenbauer[source]¶

Gegenbauer polynomial \(C_n^{\left(\alpha\right)}(x)\)

gegenbauer(n, alpha, x) gives the nth Gegenbauer polynomial in x, \(C_n^{\left(\alpha\right)}(x)\).

The Gegenbauer polynomials are orthogonal on \([-1, 1]\) with respect to the weight \(\left(1-x^2\right)^{\alpha-\frac{1}{2}}\).

Examples

>>> from sympy import gegenbauer, conjugate, diff
>>> from sympy.abc import n,a,x
>>> gegenbauer(0, a, x)
1
>>> gegenbauer(1, a, x)
2*a*x
>>> gegenbauer(2, a, x)
-a + x**2*(2*a**2 + 2*a)
>>> gegenbauer(3, a, x)
x**3*(4*a**3/3 + 4*a**2 + 8*a/3) + x*(-2*a**2 - 2*a)

>>> gegenbauer(n, a, x)
gegenbauer(n, a, x)
>>> gegenbauer(n, a, -x)
(-1)**n*gegenbauer(n, a, x)

>>> gegenbauer(n, a, 0)
2**n*sqrt(pi)*gamma(a + n/2)/(gamma(a)*gamma(1/2 - n/2)*gamma(n + 1))
>>> gegenbauer(n, a, 1)
gamma(2*a + n)/(gamma(2*a)*gamma(n + 1))

>>> conjugate(gegenbauer(n, a, x))
gegenbauer(n, conjugate(a), conjugate(x))

>>> diff(gegenbauer(n, a, x), x)
2*a*gegenbauer(n - 1, a + 1, x)

References

R388: https://en.wikipedia.org/wiki/Gegenbauer_polynomials
R389: http://mathworld.wolfram.com/GegenbauerPolynomial.html
R390: http://functions.wolfram.com/Polynomials/GegenbauerC3/

Chebyshev Polynomials¶

class sympy.functions.special.polynomials.chebyshevt[source]¶

Chebyshev polynomial of the first kind, \(T_n(x)\)

chebyshevt(n, x) gives the nth Chebyshev polynomial (of the first kind) in x, \(T_n(x)\).

The Chebyshev polynomials of the first kind are orthogonal on \([-1, 1]\) with respect to the weight \(\frac{1}{\sqrt{1-x^2}}\).

Examples

>>> from sympy import chebyshevt, chebyshevu, diff
>>> from sympy.abc import n,x
>>> chebyshevt(0, x)
1
>>> chebyshevt(1, x)
x
>>> chebyshevt(2, x)
2*x**2 - 1

>>> chebyshevt(n, x)
chebyshevt(n, x)
>>> chebyshevt(n, -x)
(-1)**n*chebyshevt(n, x)
>>> chebyshevt(-n, x)
chebyshevt(n, x)

>>> chebyshevt(n, 0)
cos(pi*n/2)
>>> chebyshevt(n, -1)
(-1)**n

>>> diff(chebyshevt(n, x), x)
n*chebyshevu(n - 1, x)

References

R391: https://en.wikipedia.org/wiki/Chebyshev_polynomial
R392: http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html
R393: http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html
R394: http://functions.wolfram.com/Polynomials/ChebyshevT/
R395: http://functions.wolfram.com/Polynomials/ChebyshevU/

class sympy.functions.special.polynomials.chebyshevu[source]¶

Chebyshev polynomial of the second kind, \(U_n(x)\)

chebyshevu(n, x) gives the nth Chebyshev polynomial of the second kind in x, \(U_n(x)\).

The Chebyshev polynomials of the second kind are orthogonal on \([-1, 1]\) with respect to the weight \(\sqrt{1-x^2}\).

Examples

>>> from sympy import chebyshevt, chebyshevu, diff
>>> from sympy.abc import n,x
>>> chebyshevu(0, x)
1
>>> chebyshevu(1, x)
2*x
>>> chebyshevu(2, x)
4*x**2 - 1

>>> chebyshevu(n, x)
chebyshevu(n, x)
>>> chebyshevu(n, -x)
(-1)**n*chebyshevu(n, x)
>>> chebyshevu(-n, x)
-chebyshevu(n - 2, x)

>>> chebyshevu(n, 0)
cos(pi*n/2)
>>> chebyshevu(n, 1)
n + 1

>>> diff(chebyshevu(n, x), x)
(-x*chebyshevu(n, x) + (n + 1)*chebyshevt(n + 1, x))/(x**2 - 1)

References

R396: https://en.wikipedia.org/wiki/Chebyshev_polynomial
R397: http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html
R398: http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html
R399: http://functions.wolfram.com/Polynomials/ChebyshevT/
R400: http://functions.wolfram.com/Polynomials/ChebyshevU/

class sympy.functions.special.polynomials.chebyshevt_root[source]¶

chebyshev_root(n, k) returns the kth root (indexed from zero) of the nth Chebyshev polynomial of the first kind; that is, if 0 <= k < n, chebyshevt(n, chebyshevt_root(n, k)) == 0.

Examples

>>> from sympy import chebyshevt, chebyshevt_root
>>> chebyshevt_root(3, 2)
-sqrt(3)/2
>>> chebyshevt(3, chebyshevt_root(3, 2))
0

class sympy.functions.special.polynomials.chebyshevu_root[source]¶

chebyshevu_root(n, k) returns the kth root (indexed from zero) of the nth Chebyshev polynomial of the second kind; that is, if 0 <= k < n, chebyshevu(n, chebyshevu_root(n, k)) == 0.

Examples

>>> from sympy import chebyshevu, chebyshevu_root
>>> chebyshevu_root(3, 2)
-sqrt(2)/2
>>> chebyshevu(3, chebyshevu_root(3, 2))
0

Legendre Polynomials¶

class sympy.functions.special.polynomials.legendre[source]¶

legendre(n, x) gives the nth Legendre polynomial of x, \(P_n(x)\)

The Legendre polynomials are orthogonal on [-1, 1] with respect to the constant weight 1. They satisfy \(P_n(1) = 1\) for all n; further, \(P_n\) is odd for odd n and even for even n.

Examples

>>> from sympy import legendre, diff
>>> from sympy.abc import x, n
>>> legendre(0, x)
1
>>> legendre(1, x)
x
>>> legendre(2, x)
3*x**2/2 - 1/2
>>> legendre(n, x)
legendre(n, x)
>>> diff(legendre(n,x), x)
n*(x*legendre(n, x) - legendre(n - 1, x))/(x**2 - 1)

References

R401: https://en.wikipedia.org/wiki/Legendre_polynomial
R402: http://mathworld.wolfram.com/LegendrePolynomial.html
R403: http://functions.wolfram.com/Polynomials/LegendreP/
R404: http://functions.wolfram.com/Polynomials/LegendreP2/

class sympy.functions.special.polynomials.assoc_legendre[source]¶

assoc_legendre(n,m, x) gives \(P_n^m(x)\), where n and m are the degree and order or an expression which is related to the nth order Legendre polynomial, \(P_n(x)\) in the following manner:

\[P_n^m(x) = (-1)^m (1 - x^2)^{\frac{m}{2}} \frac{\mathrm{d}^m P_n(x)}{\mathrm{d} x^m}\]

Associated Legendre polynomial are orthogonal on [-1, 1] with:

weight = 1 for the same m, and different n.
weight = 1/(1-x**2) for the same n, and different m.

Examples

>>> from sympy import assoc_legendre
>>> from sympy.abc import x, m, n
>>> assoc_legendre(0,0, x)
1
>>> assoc_legendre(1,0, x)
x
>>> assoc_legendre(1,1, x)
-sqrt(1 - x**2)
>>> assoc_legendre(n,m,x)
assoc_legendre(n, m, x)

References

R405: https://en.wikipedia.org/wiki/Associated_Legendre_polynomials
R406: http://mathworld.wolfram.com/LegendrePolynomial.html
R407: http://functions.wolfram.com/Polynomials/LegendreP/
R408: http://functions.wolfram.com/Polynomials/LegendreP2/

Hermite Polynomials¶

class sympy.functions.special.polynomials.hermite[source]¶

hermite(n, x) gives the nth Hermite polynomial in x, \(H_n(x)\)

The Hermite polynomials are orthogonal on \((-\infty, \infty)\) with respect to the weight \(\exp\left(-x^2\right)\).

Examples

>>> from sympy import hermite, diff
>>> from sympy.abc import x, n
>>> hermite(0, x)
1
>>> hermite(1, x)
2*x
>>> hermite(2, x)
4*x**2 - 2
>>> hermite(n, x)
hermite(n, x)
>>> diff(hermite(n,x), x)
2*n*hermite(n - 1, x)
>>> hermite(n, -x)
(-1)**n*hermite(n, x)

References

R409: https://en.wikipedia.org/wiki/Hermite_polynomial
R410: http://mathworld.wolfram.com/HermitePolynomial.html
R411: http://functions.wolfram.com/Polynomials/HermiteH/

Laguerre Polynomials¶

class sympy.functions.special.polynomials.laguerre[source]¶

Returns the nth Laguerre polynomial in x, \(L_n(x)\).

Parameters: n : int

Degree of Laguerre polynomial. Must be n >= 0.

Examples

>>> from sympy import laguerre, diff
>>> from sympy.abc import x, n
>>> laguerre(0, x)
1
>>> laguerre(1, x)
1 - x
>>> laguerre(2, x)
x**2/2 - 2*x + 1
>>> laguerre(3, x)
-x**3/6 + 3*x**2/2 - 3*x + 1

>>> laguerre(n, x)
laguerre(n, x)

>>> diff(laguerre(n, x), x)
-assoc_laguerre(n - 1, 1, x)

References

R412: https://en.wikipedia.org/wiki/Laguerre_polynomial
R413: http://mathworld.wolfram.com/LaguerrePolynomial.html
R414: http://functions.wolfram.com/Polynomials/LaguerreL/
R415: http://functions.wolfram.com/Polynomials/LaguerreL3/

class sympy.functions.special.polynomials.assoc_laguerre[source]¶

Returns the nth generalized Laguerre polynomial in x, \(L_n(x)\).

Parameters

n : int

Degree of Laguerre polynomial. Must be n >= 0.

alpha : Expr

Arbitrary expression. For alpha=0 regular Laguerre polynomials will be generated.

Examples

>>> from sympy import laguerre, assoc_laguerre, diff
>>> from sympy.abc import x, n, a
>>> assoc_laguerre(0, a, x)
1
>>> assoc_laguerre(1, a, x)
a - x + 1
>>> assoc_laguerre(2, a, x)
a**2/2 + 3*a/2 + x**2/2 + x*(-a - 2) + 1
>>> assoc_laguerre(3, a, x)
a**3/6 + a**2 + 11*a/6 - x**3/6 + x**2*(a/2 + 3/2) +
    x*(-a**2/2 - 5*a/2 - 3) + 1

>>> assoc_laguerre(n, a, 0)
binomial(a + n, a)

>>> assoc_laguerre(n, a, x)
assoc_laguerre(n, a, x)

>>> assoc_laguerre(n, 0, x)
laguerre(n, x)

>>> diff(assoc_laguerre(n, a, x), x)
-assoc_laguerre(n - 1, a + 1, x)

>>> diff(assoc_laguerre(n, a, x), a)
Sum(assoc_laguerre(_k, a, x)/(-a + n), (_k, 0, n - 1))

References

R416: https://en.wikipedia.org/wiki/Laguerre_polynomial#Assoc_laguerre_polynomials
R417: http://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html
R418: http://functions.wolfram.com/Polynomials/LaguerreL/
R419: http://functions.wolfram.com/Polynomials/LaguerreL3/

Spherical Harmonics¶

class sympy.functions.special.spherical_harmonics.Ynm[source]¶

Spherical harmonics defined as

\[Y_n^m(\theta, \varphi) := \sqrt{\frac{(2n+1)(n-m)!}{4\pi(n+m)!}} \exp(i m \varphi) \mathrm{P}_n^m\left(\cos(\theta)\right)\]

Ynm() gives the spherical harmonic function of order \(n\) and \(m\) in \(\theta\) and \(\varphi\), \(Y_n^m(\theta, \varphi)\). The four parameters are as follows: \(n \geq 0\) an integer and \(m\) an integer such that \(-n \leq m \leq n\) holds. The two angles are real-valued with \(\theta \in [0, \pi]\) and \(\varphi \in [0, 2\pi]\).

Examples

>>> from sympy import Ynm, Symbol
>>> from sympy.abc import n,m
>>> theta = Symbol("theta")
>>> phi = Symbol("phi")

>>> Ynm(n, m, theta, phi)
Ynm(n, m, theta, phi)

Several symmetries are known, for the order

>>> from sympy import Ynm, Symbol
>>> from sympy.abc import n,m
>>> theta = Symbol("theta")
>>> phi = Symbol("phi")

>>> Ynm(n, -m, theta, phi)
(-1)**m*exp(-2*I*m*phi)*Ynm(n, m, theta, phi)

as well as for the angles

>>> from sympy import Ynm, Symbol, simplify
>>> from sympy.abc import n,m
>>> theta = Symbol("theta")
>>> phi = Symbol("phi")

>>> Ynm(n, m, -theta, phi)
Ynm(n, m, theta, phi)

>>> Ynm(n, m, theta, -phi)
exp(-2*I*m*phi)*Ynm(n, m, theta, phi)

For specific integers n and m we can evaluate the harmonics to more useful expressions

>>> simplify(Ynm(0, 0, theta, phi).expand(func=True))
1/(2*sqrt(pi))

>>> simplify(Ynm(1, -1, theta, phi).expand(func=True))
sqrt(6)*exp(-I*phi)*sin(theta)/(4*sqrt(pi))

>>> simplify(Ynm(1, 0, theta, phi).expand(func=True))
sqrt(3)*cos(theta)/(2*sqrt(pi))

>>> simplify(Ynm(1, 1, theta, phi).expand(func=True))
-sqrt(6)*exp(I*phi)*sin(theta)/(4*sqrt(pi))

>>> simplify(Ynm(2, -2, theta, phi).expand(func=True))
sqrt(30)*exp(-2*I*phi)*sin(theta)**2/(8*sqrt(pi))

>>> simplify(Ynm(2, -1, theta, phi).expand(func=True))
sqrt(30)*exp(-I*phi)*sin(2*theta)/(8*sqrt(pi))

>>> simplify(Ynm(2, 0, theta, phi).expand(func=True))
sqrt(5)*(3*cos(theta)**2 - 1)/(4*sqrt(pi))

>>> simplify(Ynm(2, 1, theta, phi).expand(func=True))
-sqrt(30)*exp(I*phi)*sin(2*theta)/(8*sqrt(pi))

>>> simplify(Ynm(2, 2, theta, phi).expand(func=True))
sqrt(30)*exp(2*I*phi)*sin(theta)**2/(8*sqrt(pi))

We can differentiate the functions with respect to both angles

>>> from sympy import Ynm, Symbol, diff
>>> from sympy.abc import n,m
>>> theta = Symbol("theta")
>>> phi = Symbol("phi")

>>> diff(Ynm(n, m, theta, phi), theta)
m*cot(theta)*Ynm(n, m, theta, phi) + sqrt((-m + n)*(m + n + 1))*exp(-I*phi)*Ynm(n, m + 1, theta, phi)

>>> diff(Ynm(n, m, theta, phi), phi)
I*m*Ynm(n, m, theta, phi)

Further we can compute the complex conjugation

>>> from sympy import Ynm, Symbol, conjugate
>>> from sympy.abc import n,m
>>> theta = Symbol("theta")
>>> phi = Symbol("phi")

>>> conjugate(Ynm(n, m, theta, phi))
(-1)**(2*m)*exp(-2*I*m*phi)*Ynm(n, m, theta, phi)

To get back the well known expressions in spherical coordinates we use full expansion

>>> from sympy import Ynm, Symbol, expand_func
>>> from sympy.abc import n,m
>>> theta = Symbol("theta")
>>> phi = Symbol("phi")

>>> expand_func(Ynm(n, m, theta, phi))
sqrt((2*n + 1)*factorial(-m + n)/factorial(m + n))*exp(I*m*phi)*assoc_legendre(n, m, cos(theta))/(2*sqrt(pi))

See also

Ynm_c, Znm

References

R420: https://en.wikipedia.org/wiki/Spherical_harmonics
R421: http://mathworld.wolfram.com/SphericalHarmonic.html
R422: http://functions.wolfram.com/Polynomials/SphericalHarmonicY/
R423: http://dlmf.nist.gov/14.30

sympy.functions.special.spherical_harmonics.Ynm_c(n, m, theta, phi)[source]¶

Conjugate spherical harmonics defined as

\[\overline{Y_n^m(\theta, \varphi)} := (-1)^m Y_n^{-m}(\theta, \varphi)\]

See also

Ynm, Znm

References

R424: https://en.wikipedia.org/wiki/Spherical_harmonics
R425: http://mathworld.wolfram.com/SphericalHarmonic.html
R426: http://functions.wolfram.com/Polynomials/SphericalHarmonicY/

class sympy.functions.special.spherical_harmonics.Znm[source]¶

Real spherical harmonics defined as

\[\begin{split}Z_n^m(\theta, \varphi) := \begin{cases} \frac{Y_n^m(\theta, \varphi) + \overline{Y_n^m(\theta, \varphi)}}{\sqrt{2}} &\quad m > 0 \\ Y_n^m(\theta, \varphi) &\quad m = 0 \\ \frac{Y_n^m(\theta, \varphi) - \overline{Y_n^m(\theta, \varphi)}}{i \sqrt{2}} &\quad m < 0 \\ \end{cases}\end{split}\]

which gives in simplified form

\[\begin{split}Z_n^m(\theta, \varphi) = \begin{cases} \frac{Y_n^m(\theta, \varphi) + (-1)^m Y_n^{-m}(\theta, \varphi)}{\sqrt{2}} &\quad m > 0 \\ Y_n^m(\theta, \varphi) &\quad m = 0 \\ \frac{Y_n^m(\theta, \varphi) - (-1)^m Y_n^{-m}(\theta, \varphi)}{i \sqrt{2}} &\quad m < 0 \\ \end{cases}\end{split}\]

See also

Ynm, Ynm_c

References

R427: https://en.wikipedia.org/wiki/Spherical_harmonics
R428: http://mathworld.wolfram.com/SphericalHarmonic.html
R429: http://functions.wolfram.com/Polynomials/SphericalHarmonicY/

Tensor Functions¶

sympy.functions.special.tensor_functions.Eijk(*args, **kwargs)[source]¶

Represent the Levi-Civita symbol.

This is just compatibility wrapper to LeviCivita().

See also

LeviCivita

sympy.functions.special.tensor_functions.eval_levicivita(*args)[source]¶: Evaluate Levi-Civita symbol.

class sympy.functions.special.tensor_functions.LeviCivita[source]¶

Represent the Levi-Civita symbol.

For even permutations of indices it returns 1, for odd permutations -1, and for everything else (a repeated index) it returns 0.

Thus it represents an alternating pseudotensor.

Examples

>>> from sympy import LeviCivita
>>> from sympy.abc import i, j, k
>>> LeviCivita(1, 2, 3)
1
>>> LeviCivita(1, 3, 2)
-1
>>> LeviCivita(1, 2, 2)
0
>>> LeviCivita(i, j, k)
LeviCivita(i, j, k)
>>> LeviCivita(i, j, i)
0

See also

Eijk

class sympy.functions.special.tensor_functions.KroneckerDelta[source]¶

The discrete, or Kronecker, delta function.

A function that takes in two integers \(i\) and \(j\). It returns \(0\) if \(i\) and \(j\) are not equal or it returns \(1\) if \(i\) and \(j\) are equal.

Parameters

i : Number, Symbol

The first index of the delta function.

j : Number, Symbol

The second index of the delta function.

Examples

A simple example with integer indices:

>>> from sympy.functions.special.tensor_functions import KroneckerDelta
>>> KroneckerDelta(1, 2)
0
>>> KroneckerDelta(3, 3)
1

Symbolic indices:

>>> from sympy.abc import i, j, k
>>> KroneckerDelta(i, j)
KroneckerDelta(i, j)
>>> KroneckerDelta(i, i)
1
>>> KroneckerDelta(i, i + 1)
0
>>> KroneckerDelta(i, i + 1 + k)
KroneckerDelta(i, i + k + 1)

References

R430: https://en.wikipedia.org/wiki/Kronecker_delta

classmethod eval(i, j)[source]¶

Evaluates the discrete delta function.

Examples

>>> from sympy.functions.special.tensor_functions import KroneckerDelta
>>> from sympy.abc import i, j, k

>>> KroneckerDelta(i, j)
KroneckerDelta(i, j)
>>> KroneckerDelta(i, i)
1
>>> KroneckerDelta(i, i + 1)
0
>>> KroneckerDelta(i, i + 1 + k)
KroneckerDelta(i, i + k + 1)

# indirect doctest

indices_contain_equal_information¶

Returns True if indices are either both above or below fermi.

Examples

>>> from sympy.functions.special.tensor_functions import KroneckerDelta
>>> from sympy import Symbol
>>> a = Symbol('a', above_fermi=True)
>>> i = Symbol('i', below_fermi=True)
>>> p = Symbol('p')
>>> q = Symbol('q')
>>> KroneckerDelta(p, q).indices_contain_equal_information
True
>>> KroneckerDelta(p, q+1).indices_contain_equal_information
True
>>> KroneckerDelta(i, p).indices_contain_equal_information
False

is_above_fermi¶

True if Delta can be non-zero above fermi

Examples

>>> from sympy.functions.special.tensor_functions import KroneckerDelta
>>> from sympy import Symbol
>>> a = Symbol('a', above_fermi=True)
>>> i = Symbol('i', below_fermi=True)
>>> p = Symbol('p')
>>> q = Symbol('q')
>>> KroneckerDelta(p, a).is_above_fermi
True
>>> KroneckerDelta(p, i).is_above_fermi
False
>>> KroneckerDelta(p, q).is_above_fermi
True

is_below_fermi¶

True if Delta can be non-zero below fermi

Examples

>>> from sympy.functions.special.tensor_functions import KroneckerDelta
>>> from sympy import Symbol
>>> a = Symbol('a', above_fermi=True)
>>> i = Symbol('i', below_fermi=True)
>>> p = Symbol('p')
>>> q = Symbol('q')
>>> KroneckerDelta(p, a).is_below_fermi
False
>>> KroneckerDelta(p, i).is_below_fermi
True
>>> KroneckerDelta(p, q).is_below_fermi
True

is_only_above_fermi¶

True if Delta is restricted to above fermi

Examples

>>> from sympy.functions.special.tensor_functions import KroneckerDelta
>>> from sympy import Symbol
>>> a = Symbol('a', above_fermi=True)
>>> i = Symbol('i', below_fermi=True)
>>> p = Symbol('p')
>>> q = Symbol('q')
>>> KroneckerDelta(p, a).is_only_above_fermi
True
>>> KroneckerDelta(p, q).is_only_above_fermi
False
>>> KroneckerDelta(p, i).is_only_above_fermi
False

is_only_below_fermi¶

True if Delta is restricted to below fermi

Examples

>>> from sympy.functions.special.tensor_functions import KroneckerDelta
>>> from sympy import Symbol
>>> a = Symbol('a', above_fermi=True)
>>> i = Symbol('i', below_fermi=True)
>>> p = Symbol('p')
>>> q = Symbol('q')
>>> KroneckerDelta(p, i).is_only_below_fermi
True
>>> KroneckerDelta(p, q).is_only_below_fermi
False
>>> KroneckerDelta(p, a).is_only_below_fermi
False

killable_index¶

Returns the index which is preferred to substitute in the final expression.

The index to substitute is the index with less information regarding fermi level. If indices contain same information, ‘a’ is preferred before ‘b’.

Examples

>>> from sympy.functions.special.tensor_functions import KroneckerDelta
>>> from sympy import Symbol
>>> a = Symbol('a', above_fermi=True)
>>> i = Symbol('i', below_fermi=True)
>>> j = Symbol('j', below_fermi=True)
>>> p = Symbol('p')
>>> KroneckerDelta(p, i).killable_index
p
>>> KroneckerDelta(p, a).killable_index
p
>>> KroneckerDelta(i, j).killable_index
j

See also

preferred_index

preferred_index¶

Returns the index which is preferred to keep in the final expression.

The preferred index is the index with more information regarding fermi level. If indices contain same information, ‘a’ is preferred before ‘b’.

Examples

>>> from sympy.functions.special.tensor_functions import KroneckerDelta
>>> from sympy import Symbol
>>> a = Symbol('a', above_fermi=True)
>>> i = Symbol('i', below_fermi=True)
>>> j = Symbol('j', below_fermi=True)
>>> p = Symbol('p')
>>> KroneckerDelta(p, i).preferred_index
i
>>> KroneckerDelta(p, a).preferred_index
a
>>> KroneckerDelta(i, j).preferred_index
i

See also

killable_index

Special¶

DiracDelta¶

Heaviside¶

Singularity Function¶

Gamma, Beta and related Functions¶

Error Functions and Fresnel Integrals¶

Exponential, Logarithmic and Trigonometric Integrals¶

Bessel Type Functions¶

Airy Functions¶

B-Splines¶

Riemann Zeta and Related Functions¶

Hypergeometric Functions¶

Elliptic integrals¶

Mathieu Functions¶

Orthogonal Polynomials¶

Jacobi Polynomials¶

Gegenbauer Polynomials¶

Chebyshev Polynomials¶

Legendre Polynomials¶

Hermite Polynomials¶

Laguerre Polynomials¶

Spherical Harmonics¶

Tensor Functions¶

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