Elementary¶

This module implements elementary functions such as trigonometric, hyperbolic, and sqrt, as well as functions like Abs, Max, Min etc.

sympy.functions.elementary.complexes¶

re¶

class sympy.functions.elementary.complexes.re[source]¶

Returns real part of expression. This function performs only elementary analysis and so it will fail to decompose properly more complicated expressions. If completely simplified result is needed then use Basic.as_real_imag() or perform complex expansion on instance of this function.

Examples

>>> from sympy import re, im, I, E
>>> from sympy.abc import x, y
>>> re(2*E)
2*E
>>> re(2*I + 17)
17
>>> re(2*I)
0
>>> re(im(x) + x*I + 2)
2

See also

im

as_real_imag(deep=True, **hints)[source]¶: Returns the real number with a zero imaginary part.

im¶

class sympy.functions.elementary.complexes.im[source]¶

Returns imaginary part of expression. This function performs only elementary analysis and so it will fail to decompose properly more complicated expressions. If completely simplified result is needed then use Basic.as_real_imag() or perform complex expansion on instance of this function.

Examples

>>> from sympy import re, im, E, I
>>> from sympy.abc import x, y
>>> im(2*E)
0
>>> re(2*I + 17)
17
>>> im(x*I)
re(x)
>>> im(re(x) + y)
im(y)

See also

re

as_real_imag(deep=True, **hints)[source]¶

Return the imaginary part with a zero real part.

Examples

>>> from sympy.functions import im
>>> from sympy import I
>>> im(2 + 3*I).as_real_imag()
(3, 0)

sign¶

class sympy.functions.elementary.complexes.sign[source]¶

Returns the complex sign of an expression:

If the expression is real the sign will be:

1 if expression is positive

0 if expression is equal to zero

-1 if expression is negative

If the expression is imaginary the sign will be:

I if im(expression) is positive

-I if im(expression) is negative

Otherwise an unevaluated expression will be returned. When evaluated, the result (in general) will be cos(arg(expr)) + I*sin(arg(expr)).

Examples

>>> from sympy.functions import sign
>>> from sympy.core.numbers import I

>>> sign(-1)
-1
>>> sign(0)
0
>>> sign(-3*I)
-I
>>> sign(1 + I)
sign(1 + I)
>>> _.evalf()
0.707106781186548 + 0.707106781186548*I

See also

Abs, conjugate

Abs¶

class sympy.functions.elementary.complexes.Abs[source]¶

Return the absolute value of the argument.

This is an extension of the built-in function abs() to accept symbolic values. If you pass a SymPy expression to the built-in abs(), it will pass it automatically to Abs().

Examples

>>> from sympy import Abs, Symbol, S
>>> Abs(-1)
1
>>> x = Symbol('x', real=True)
>>> Abs(-x)
Abs(x)
>>> Abs(x**2)
x**2
>>> abs(-x) # The Python built-in
Abs(x)

Note that the Python built-in will return either an Expr or int depending on the argument:

>>> type(abs(-1))
<... 'int'>
>>> type(abs(S.NegativeOne))
<class 'sympy.core.numbers.One'>

Abs will always return a sympy object.

See also

sign, conjugate

fdiff(argindex=1)[source]¶

Get the first derivative of the argument to Abs().

Examples

>>> from sympy.abc import x
>>> from sympy.functions import Abs
>>> Abs(-x).fdiff()
sign(x)

arg¶

class sympy.functions.elementary.complexes.arg[source]¶

Returns the argument (in radians) of a complex number. For a positive number, the argument is always 0.

Examples

>>> from sympy.functions import arg
>>> from sympy import I, sqrt
>>> arg(2.0)
0
>>> arg(I)
pi/2
>>> arg(sqrt(2) + I*sqrt(2))
pi/4

conjugate¶

class sympy.functions.elementary.complexes.conjugate[source]¶

Returns the \(complex conjugate\) Ref[1] of an argument. In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part.

Thus, the conjugate of the complex number \(a + ib\) (where a and b are real numbers) is \(a - ib\)

Examples

>>> from sympy import conjugate, I
>>> conjugate(2)
2
>>> conjugate(I)
-I

See also

sign, Abs

References

R176: https://en.wikipedia.org/wiki/Complex_conjugation

polar_lift¶

class sympy.functions.elementary.complexes.polar_lift[source]¶

Lift argument to the Riemann surface of the logarithm, using the standard branch.

>>> from sympy import Symbol, polar_lift, I
>>> p = Symbol('p', polar=True)
>>> x = Symbol('x')
>>> polar_lift(4)
4*exp_polar(0)
>>> polar_lift(-4)
4*exp_polar(I*pi)
>>> polar_lift(-I)
exp_polar(-I*pi/2)
>>> polar_lift(I + 2)
polar_lift(2 + I)

>>> polar_lift(4*x)
4*polar_lift(x)
>>> polar_lift(4*p)
4*p

See also

sympy.functions.elementary.exponential.exp_polar, periodic_argument

periodic_argument¶

class sympy.functions.elementary.complexes.periodic_argument[source]¶

Represent the argument on a quotient of the Riemann surface of the logarithm. That is, given a period P, always return a value in (-P/2, P/2], by using exp(P*I) == 1.

>>> from sympy import exp, exp_polar, periodic_argument, unbranched_argument
>>> from sympy import I, pi
>>> unbranched_argument(exp(5*I*pi))
pi
>>> unbranched_argument(exp_polar(5*I*pi))
5*pi
>>> periodic_argument(exp_polar(5*I*pi), 2*pi)
pi
>>> periodic_argument(exp_polar(5*I*pi), 3*pi)
-pi
>>> periodic_argument(exp_polar(5*I*pi), pi)
0

See also

sympy.functions.elementary.exponential.exp_polar

polar_lift: Lift argument to the Riemann surface of the logarithm

principal_branch

principal_branch¶

class sympy.functions.elementary.complexes.principal_branch[source]¶

Represent a polar number reduced to its principal branch on a quotient of the Riemann surface of the logarithm.

This is a function of two arguments. The first argument is a polar number \(z\), and the second one a positive real number of infinity, \(p\). The result is “z mod exp_polar(I*p)”.

>>> from sympy import exp_polar, principal_branch, oo, I, pi
>>> from sympy.abc import z
>>> principal_branch(z, oo)
z
>>> principal_branch(exp_polar(2*pi*I)*3, 2*pi)
3*exp_polar(0)
>>> principal_branch(exp_polar(2*pi*I)*3*z, 2*pi)
3*principal_branch(z, 2*pi)

See also

sympy.functions.elementary.exponential.exp_polar

polar_lift: Lift argument to the Riemann surface of the logarithm

periodic_argument

sympy.functions.elementary.trigonometric¶

Trigonometric Functions¶

sin¶

class sympy.functions.elementary.trigonometric.sin[source]¶

The sine function.

Returns the sine of x (measured in radians).

Notes

This function will evaluate automatically in the case x/pi is some rational number [R180]. For example, if x is a multiple of pi, pi/2, pi/3, pi/4 and pi/6.

Examples

>>> from sympy import sin, pi
>>> from sympy.abc import x
>>> sin(x**2).diff(x)
2*x*cos(x**2)
>>> sin(1).diff(x)
0
>>> sin(pi)
0
>>> sin(pi/2)
1
>>> sin(pi/6)
1/2
>>> sin(pi/12)
-sqrt(2)/4 + sqrt(6)/4

References

R177: https://en.wikipedia.org/wiki/Trigonometric_functions
R178: http://dlmf.nist.gov/4.14
R179: http://functions.wolfram.com/ElementaryFunctions/Sin
R180(1,2): http://mathworld.wolfram.com/TrigonometryAngles.html

Members

cos¶

class sympy.functions.elementary.trigonometric.cos[source]¶

The cosine function.

Returns the cosine of x (measured in radians).

Notes

See sin() for notes about automatic evaluation.

Examples

>>> from sympy import cos, pi
>>> from sympy.abc import x
>>> cos(x**2).diff(x)
-2*x*sin(x**2)
>>> cos(1).diff(x)
0
>>> cos(pi)
-1
>>> cos(pi/2)
0
>>> cos(2*pi/3)
-1/2
>>> cos(pi/12)
sqrt(2)/4 + sqrt(6)/4

References

R181: https://en.wikipedia.org/wiki/Trigonometric_functions
R182: http://dlmf.nist.gov/4.14
R183: http://functions.wolfram.com/ElementaryFunctions/Cos

Members

tan¶

class sympy.functions.elementary.trigonometric.tan[source]¶

The tangent function.

Returns the tangent of x (measured in radians).

Notes

See sin() for notes about automatic evaluation.

Examples

>>> from sympy import tan, pi
>>> from sympy.abc import x
>>> tan(x**2).diff(x)
2*x*(tan(x**2)**2 + 1)
>>> tan(1).diff(x)
0
>>> tan(pi/8).expand()
-1 + sqrt(2)

References

R184: https://en.wikipedia.org/wiki/Trigonometric_functions
R185: http://dlmf.nist.gov/4.14
R186: http://functions.wolfram.com/ElementaryFunctions/Tan

Members

cot¶

class sympy.functions.elementary.trigonometric.cot[source]¶

The cotangent function.

Returns the cotangent of x (measured in radians).

Notes

See sin() for notes about automatic evaluation.

Examples

>>> from sympy import cot, pi
>>> from sympy.abc import x
>>> cot(x**2).diff(x)
2*x*(-cot(x**2)**2 - 1)
>>> cot(1).diff(x)
0
>>> cot(pi/12)
sqrt(3) + 2

References

R187: https://en.wikipedia.org/wiki/Trigonometric_functions
R188: http://dlmf.nist.gov/4.14
R189: http://functions.wolfram.com/ElementaryFunctions/Cot

Members

sec¶

class sympy.functions.elementary.trigonometric.sec[source]¶

The secant function.

Returns the secant of x (measured in radians).

Notes

See sin() for notes about automatic evaluation.

Examples

>>> from sympy import sec
>>> from sympy.abc import x
>>> sec(x**2).diff(x)
2*x*tan(x**2)*sec(x**2)
>>> sec(1).diff(x)
0

References

R190: https://en.wikipedia.org/wiki/Trigonometric_functions
R191: http://dlmf.nist.gov/4.14
R192: http://functions.wolfram.com/ElementaryFunctions/Sec

Members

csc¶

class sympy.functions.elementary.trigonometric.csc[source]¶

The cosecant function.

Returns the cosecant of x (measured in radians).

Notes

See sin() for notes about automatic evaluation.

Examples

>>> from sympy import csc
>>> from sympy.abc import x
>>> csc(x**2).diff(x)
-2*x*cot(x**2)*csc(x**2)
>>> csc(1).diff(x)
0

References

R193: https://en.wikipedia.org/wiki/Trigonometric_functions
R194: http://dlmf.nist.gov/4.14
R195: http://functions.wolfram.com/ElementaryFunctions/Csc

Members

sinc¶

class sympy.functions.elementary.trigonometric.sinc[source]¶

Represents unnormalized sinc function

Examples

>>> from sympy import sinc, oo, jn, Product, Symbol
>>> from sympy.abc import x
>>> sinc(x)
sinc(x)

Automated Evaluation

>>> sinc(0)
1
>>> sinc(oo)
0

Differentiation

>>> sinc(x).diff()
(x*cos(x) - sin(x))/x**2

Series Expansion

>>> sinc(x).series()
1 - x**2/6 + x**4/120 + O(x**6)

As zero’th order spherical Bessel Function

>>> sinc(x).rewrite(jn)
jn(0, x)

References

R196: https://en.wikipedia.org/wiki/Sinc_function

Members

Trigonometric Inverses¶

asin¶

class sympy.functions.elementary.trigonometric.asin[source]¶

The inverse sine function.

Returns the arcsine of x in radians.

Notes

asin(x) will evaluate automatically in the cases oo, -oo, 0, 1, -1 and for some instances when the result is a rational multiple of pi (see the eval class method).

Examples

>>> from sympy import asin, oo, pi
>>> asin(1)
pi/2
>>> asin(-1)
-pi/2

References

R197: https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
R198: http://dlmf.nist.gov/4.23
R199: http://functions.wolfram.com/ElementaryFunctions/ArcSin

Members

acos¶

class sympy.functions.elementary.trigonometric.acos[source]¶

The inverse cosine function.

Returns the arc cosine of x (measured in radians).

Notes

acos(x) will evaluate automatically in the cases oo, -oo, 0, 1, -1.

acos(zoo) evaluates to zoo (see note in :py:class`sympy.functions.elementary.trigonometric.asec`)

Examples

>>> from sympy import acos, oo, pi
>>> acos(1)
0
>>> acos(0)
pi/2
>>> acos(oo)
oo*I

References

R200: https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
R201: http://dlmf.nist.gov/4.23
R202: http://functions.wolfram.com/ElementaryFunctions/ArcCos

Members

atan¶

class sympy.functions.elementary.trigonometric.atan[source]¶

The inverse tangent function.

Returns the arc tangent of x (measured in radians).

Notes

atan(x) will evaluate automatically in the cases oo, -oo, 0, 1, -1.

Examples

>>> from sympy import atan, oo, pi
>>> atan(0)
0
>>> atan(1)
pi/4
>>> atan(oo)
pi/2

References

R203: https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
R204: http://dlmf.nist.gov/4.23
R205: http://functions.wolfram.com/ElementaryFunctions/ArcTan

Members

acot¶

class sympy.functions.elementary.trigonometric.acot[source]¶

The inverse cotangent function.

Returns the arc cotangent of x (measured in radians).

References

R206: https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
R207: http://dlmf.nist.gov/4.23
R208: http://functions.wolfram.com/ElementaryFunctions/ArcCot

Members

asec¶

class sympy.functions.elementary.trigonometric.asec[source]¶

The inverse secant function.

Returns the arc secant of x (measured in radians).

Notes

asec(x) will evaluate automatically in the cases oo, -oo, 0, 1, -1.

asec(x) has branch cut in the interval [-1, 1]. For complex arguments, it can be defined [R212] as

\[sec^{-1}(z) = -i*(log(\sqrt{1 - z^2} + 1) / z)\]

At x = 0, for positive branch cut, the limit evaluates to zoo. For negative branch cut, the limit

\[\lim_{z \to 0}-i*(log(-\sqrt{1 - z^2} + 1) / z)\]

simplifies to \(-i*log(z/2 + O(z^3))\) which ultimately evaluates to zoo.

As asex(x) = asec(1/x), a similar argument can be given for acos(x).

Examples

>>> from sympy import asec, oo, pi
>>> asec(1)
0
>>> asec(-1)
pi

References

R209: https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
R210: http://dlmf.nist.gov/4.23
R211: http://functions.wolfram.com/ElementaryFunctions/ArcSec
R212(1,2): http://reference.wolfram.com/language/ref/ArcSec.html

Members

acsc¶

class sympy.functions.elementary.trigonometric.acsc[source]¶

The inverse cosecant function.

Returns the arc cosecant of x (measured in radians).

Notes

acsc(x) will evaluate automatically in the cases oo, -oo, 0, 1, -1.

Examples

>>> from sympy import acsc, oo, pi
>>> acsc(1)
pi/2
>>> acsc(-1)
-pi/2

References

R213: https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
R214: http://dlmf.nist.gov/4.23
R215: http://functions.wolfram.com/ElementaryFunctions/ArcCsc

Members

atan2¶

class sympy.functions.elementary.trigonometric.atan2[source]¶

The function atan2(y, x) computes \(\operatorname{atan}(y/x)\) taking two arguments \(y\) and \(x\). Signs of both \(y\) and \(x\) are considered to determine the appropriate quadrant of \(\operatorname{atan}(y/x)\). The range is \((-\pi, \pi]\). The complete definition reads as follows:

\[\begin{split}\operatorname{atan2}(y, x) = \begin{cases} \arctan\left(\frac y x\right) & \qquad x > 0 \\ \arctan\left(\frac y x\right) + \pi& \qquad y \ge 0 , x < 0 \\ \arctan\left(\frac y x\right) - \pi& \qquad y < 0 , x < 0 \\ +\frac{\pi}{2} & \qquad y > 0 , x = 0 \\ -\frac{\pi}{2} & \qquad y < 0 , x = 0 \\ \text{undefined} & \qquad y = 0, x = 0 \end{cases}\end{split}\]

Attention: Note the role reversal of both arguments. The \(y\)-coordinate is the first argument and the \(x\)-coordinate the second.

Examples

Going counter-clock wise around the origin we find the following angles:

>>> from sympy import atan2
>>> atan2(0, 1)
0
>>> atan2(1, 1)
pi/4
>>> atan2(1, 0)
pi/2
>>> atan2(1, -1)
3*pi/4
>>> atan2(0, -1)
pi
>>> atan2(-1, -1)
-3*pi/4
>>> atan2(-1, 0)
-pi/2
>>> atan2(-1, 1)
-pi/4

which are all correct. Compare this to the results of the ordinary \(\operatorname{atan}\) function for the point \((x, y) = (-1, 1)\)

>>> from sympy import atan, S
>>> atan(S(1) / -1)
-pi/4
>>> atan2(1, -1)
3*pi/4

where only the \(\operatorname{atan2}\) function reurns what we expect. We can differentiate the function with respect to both arguments:

>>> from sympy import diff
>>> from sympy.abc import x, y
>>> diff(atan2(y, x), x)
-y/(x**2 + y**2)

>>> diff(atan2(y, x), y)
x/(x**2 + y**2)

We can express the \(\operatorname{atan2}\) function in terms of complex logarithms:

>>> from sympy import log
>>> atan2(y, x).rewrite(log)
-I*log((x + I*y)/sqrt(x**2 + y**2))

and in terms of \(\operatorname(atan)\):

>>> from sympy import atan
>>> atan2(y, x).rewrite(atan)
2*atan(y/(x + sqrt(x**2 + y**2)))

but note that this form is undefined on the negative real axis.

References

R216: https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
R217: https://en.wikipedia.org/wiki/Atan2
R218: http://functions.wolfram.com/ElementaryFunctions/ArcTan2

Members

sympy.functions.elementary.hyperbolic¶

Hyperbolic Functions¶

HyperbolicFunction¶

class sympy.functions.elementary.hyperbolic.HyperbolicFunction[source]¶

Base class for hyperbolic functions.

See also

sinh, cosh, tanh, coth

Members

sinh¶

class sympy.functions.elementary.hyperbolic.sinh[source]¶

The hyperbolic sine function, \(\frac{e^x - e^{-x}}{2}\).

sinh(x) -> Returns the hyperbolic sine of x

See also

cosh, tanh, asinh

Members

cosh¶

class sympy.functions.elementary.hyperbolic.cosh[source]¶

The hyperbolic cosine function, \(\frac{e^x + e^{-x}}{2}\).

cosh(x) -> Returns the hyperbolic cosine of x

See also

sinh, tanh, acosh

Members

tanh¶

class sympy.functions.elementary.hyperbolic.tanh[source]¶

The hyperbolic tangent function, \(\frac{\sinh(x)}{\cosh(x)}\).

tanh(x) -> Returns the hyperbolic tangent of x

See also

sinh, cosh, atanh

Members

coth¶

class sympy.functions.elementary.hyperbolic.coth[source]¶

The hyperbolic cotangent function, \(\frac{\cosh(x)}{\sinh(x)}\).

coth(x) -> Returns the hyperbolic cotangent of x

Members

sech¶

class sympy.functions.elementary.hyperbolic.sech[source]¶

The hyperbolic secant function, \(\frac{2}{e^x + e^{-x}}\)

sech(x) -> Returns the hyperbolic secant of x

Members

csch¶

class sympy.functions.elementary.hyperbolic.csch[source]¶

The hyperbolic cosecant function, \(\frac{2}{e^x - e^{-x}}\)

csch(x) -> Returns the hyperbolic cosecant of x

Members

Hyperbolic Inverses¶

asinh¶

class sympy.functions.elementary.hyperbolic.asinh[source]¶

The inverse hyperbolic sine function.

asinh(x) -> Returns the inverse hyperbolic sine of x

See also

acosh, atanh, sinh

Members

acosh¶

class sympy.functions.elementary.hyperbolic.acosh[source]¶

The inverse hyperbolic cosine function.

acosh(x) -> Returns the inverse hyperbolic cosine of x

See also

asinh, atanh, cosh

Members

atanh¶

class sympy.functions.elementary.hyperbolic.atanh[source]¶

The inverse hyperbolic tangent function.

atanh(x) -> Returns the inverse hyperbolic tangent of x

See also

asinh, acosh, tanh

Members

acoth¶

class sympy.functions.elementary.hyperbolic.acoth[source]¶

The inverse hyperbolic cotangent function.

acoth(x) -> Returns the inverse hyperbolic cotangent of x

Members

asech¶

class sympy.functions.elementary.hyperbolic.asech[source]¶

The inverse hyperbolic secant function.

asech(x) -> Returns the inverse hyperbolic secant of x

Examples

>>> from sympy import asech, sqrt, S
>>> from sympy.abc import x
>>> asech(x).diff(x)
-1/(x*sqrt(1 - x**2))
>>> asech(1).diff(x)
0
>>> asech(1)
0
>>> asech(S(2))
I*pi/3
>>> asech(-sqrt(2))
3*I*pi/4
>>> asech((sqrt(6) - sqrt(2)))
I*pi/12

See also

asinh, atanh, cosh, acoth

References

R219: https://en.wikipedia.org/wiki/Hyperbolic_function
R220: http://dlmf.nist.gov/4.37
R221: http://functions.wolfram.com/ElementaryFunctions/ArcSech/

Members

acsch¶

class sympy.functions.elementary.hyperbolic.acsch[source]¶

The inverse hyperbolic cosecant function.

acsch(x) -> Returns the inverse hyperbolic cosecant of x

Examples

>>> from sympy import acsch, sqrt, S
>>> from sympy.abc import x
>>> acsch(x).diff(x)
-1/(x**2*sqrt(1 + x**(-2)))
>>> acsch(1).diff(x)
0
>>> acsch(1)
log(1 + sqrt(2))
>>> acsch(S.ImaginaryUnit)
-I*pi/2
>>> acsch(-2*S.ImaginaryUnit)
I*pi/6
>>> acsch(S.ImaginaryUnit*(sqrt(6) - sqrt(2)))
-5*I*pi/12

References

R222: https://en.wikipedia.org/wiki/Hyperbolic_function
R223: http://dlmf.nist.gov/4.37
R224: http://functions.wolfram.com/ElementaryFunctions/ArcCsch/

Members

sympy.functions.elementary.integers¶

ceiling¶

class sympy.functions.elementary.integers.ceiling[source]¶

Ceiling is a univariate function which returns the smallest integer value not less than its argument. This implementation generalizes ceiling to complex numbers by taking the ceiling of the real and imaginary parts separately.

Examples

>>> from sympy import ceiling, E, I, S, Float, Rational
>>> ceiling(17)
17
>>> ceiling(Rational(23, 10))
3
>>> ceiling(2*E)
6
>>> ceiling(-Float(0.567))
0
>>> ceiling(I/2)
I
>>> ceiling(S(5)/2 + 5*I/2)
3 + 3*I

References

R225: “Concrete mathematics” by Graham, pp. 87
R226: http://mathworld.wolfram.com/CeilingFunction.html

Members

floor¶

class sympy.functions.elementary.integers.floor[source]¶

Floor is a univariate function which returns the largest integer value not greater than its argument. This implementation generalizes floor to complex numbers by taking the floor of the real and imaginary parts separately.

Examples

>>> from sympy import floor, E, I, S, Float, Rational
>>> floor(17)
17
>>> floor(Rational(23, 10))
2
>>> floor(2*E)
5
>>> floor(-Float(0.567))
-1
>>> floor(-I/2)
-I
>>> floor(S(5)/2 + 5*I/2)
2 + 2*I

References

R227: “Concrete mathematics” by Graham, pp. 87
R228: http://mathworld.wolfram.com/FloorFunction.html

Members

RoundFunction¶

class sympy.functions.elementary.integers.RoundFunction[source]¶: The base class for rounding functions.

frac¶

class sympy.functions.elementary.integers.frac[source]¶

Represents the fractional part of x

For real numbers it is defined [R229] as

\[x - \left\lfloor{x}\right\rfloor\]

Examples

>>> from sympy import Symbol, frac, Rational, floor, ceiling, I
>>> frac(Rational(4, 3))
1/3
>>> frac(-Rational(4, 3))
2/3

returns zero for integer arguments

>>> n = Symbol('n', integer=True)
>>> frac(n)
0

rewrite as floor

>>> x = Symbol('x')
>>> frac(x).rewrite(floor)
x - floor(x)

for complex arguments

>>> r = Symbol('r', real=True)
>>> t = Symbol('t', real=True)
>>> frac(t + I*r)
I*frac(r) + frac(t)

References

R229(1,2): https://en.wikipedia.org/wiki/Fractional_part
R230: http://mathworld.wolfram.com/FractionalPart.html

sympy.functions.elementary.exponential¶

exp¶

class sympy.functions.elementary.exponential.exp[source]¶

The exponential function, \(e^x\).

See also

log

Members

LambertW¶

class sympy.functions.elementary.exponential.LambertW[source]¶

The Lambert W function \(W(z)\) is defined as the inverse function of \(w \exp(w)\) [R231].

In other words, the value of \(W(z)\) is such that \(z = W(z) \exp(W(z))\) for any complex number \(z\). The Lambert W function is a multivalued function with infinitely many branches \(W_k(z)\), indexed by \(k \in \mathbb{Z}\). Each branch gives a different solution \(w\) of the equation \(z = w \exp(w)\).

The Lambert W function has two partially real branches: the principal branch (\(k = 0\)) is real for real \(z > -1/e\), and the \(k = -1\) branch is real for \(-1/e < z < 0\). All branches except \(k = 0\) have a logarithmic singularity at \(z = 0\).

Examples

>>> from sympy import LambertW
>>> LambertW(1.2)
0.635564016364870
>>> LambertW(1.2, -1).n()
-1.34747534407696 - 4.41624341514535*I
>>> LambertW(-1).is_real
False

References

R231(1,2): https://en.wikipedia.org/wiki/Lambert_W_function

Members

log¶

class sympy.functions.elementary.exponential.log[source]¶

The natural logarithm function \(\ln(x)\) or \(\log(x)\). Logarithms are taken with the natural base, \(e\). To get a logarithm of a different base b, use log(x, b), which is essentially short-hand for log(x)/log(b).

See also

exp

Members

sympy.functions.elementary.piecewise¶

ExprCondPair¶

class sympy.functions.elementary.piecewise.ExprCondPair[source]¶

Represents an expression, condition pair.

Members

Piecewise¶

class sympy.functions.elementary.piecewise.Piecewise[source]¶

Represents a piecewise function.

Usage:

Piecewise( (expr,cond), (expr,cond), … )

Each argument is a 2-tuple defining an expression and condition

The conds are evaluated in turn returning the first that is True. If any of the evaluated conds are not determined explicitly False, e.g. x < 1, the function is returned in symbolic form.

If the function is evaluated at a place where all conditions are False, nan will be returned.

Pairs where the cond is explicitly False, will be removed.

Examples

>>> from sympy import Piecewise, log, ITE, piecewise_fold
>>> from sympy.abc import x, y
>>> f = x**2
>>> g = log(x)
>>> p = Piecewise((0, x < -1), (f, x <= 1), (g, True))
>>> p.subs(x,1)
1
>>> p.subs(x,5)
log(5)

Booleans can contain Piecewise elements:

>>> cond = (x < y).subs(x, Piecewise((2, x < 0), (3, True))); cond
Piecewise((2, x < 0), (3, True)) < y

The folded version of this results in a Piecewise whose expressions are Booleans:

>>> folded_cond = piecewise_fold(cond); folded_cond
Piecewise((2 < y, x < 0), (3 < y, True))

When a Boolean containing Piecewise (like cond) or a Piecewise with Boolean expressions (like folded_cond) is used as a condition, it is converted to an equivalent ITE object:

>>> Piecewise((1, folded_cond))
Piecewise((1, ITE(x < 0, y > 2, y > 3)))

When a condition is an ITE, it will be converted to a simplified Boolean expression:

>>> piecewise_fold(_)
Piecewise((1, ((x >= 0) | (y > 2)) & ((y > 3) | (x < 0))))

See also

piecewise_fold, ITE

Members

sympy.functions.elementary.piecewise.piecewise_fold(expr)[source]¶

Takes an expression containing a piecewise function and returns the expression in piecewise form. In addition, any ITE conditions are rewritten in negation normal form and simplified.

Examples

>>> from sympy import Piecewise, piecewise_fold, sympify as S
>>> from sympy.abc import x
>>> p = Piecewise((x, x < 1), (1, S(1) <= x))
>>> piecewise_fold(x*p)
Piecewise((x**2, x < 1), (x, True))

See also

Piecewise

sympy.functions.elementary.miscellaneous¶

IdentityFunction¶

class sympy.functions.elementary.miscellaneous.IdentityFunction[source]¶

The identity function

Examples

>>> from sympy import Id, Symbol
>>> x = Symbol('x')
>>> Id(x)
x

Members

Min¶

class sympy.functions.elementary.miscellaneous.Min[source]¶

Return, if possible, the minimum value of the list. It is named Min and not min to avoid conflicts with the built-in function min.

Examples

>>> from sympy import Min, Symbol, oo
>>> from sympy.abc import x, y
>>> p = Symbol('p', positive=True)
>>> n = Symbol('n', negative=True)

>>> Min(x, -2)                  #doctest: +SKIP
Min(x, -2)
>>> Min(x, -2).subs(x, 3)
-2
>>> Min(p, -3)
-3
>>> Min(x, y)                   #doctest: +SKIP
Min(x, y)
>>> Min(n, 8, p, -7, p, oo)     #doctest: +SKIP
Min(n, -7)

See also

Max: find maximum values

Members

Max¶

class sympy.functions.elementary.miscellaneous.Max[source]¶

Return, if possible, the maximum value of the list.

When number of arguments is equal one, then return this argument.

When number of arguments is equal two, then return, if possible, the value from (a, b) that is >= the other.

In common case, when the length of list greater than 2, the task is more complicated. Return only the arguments, which are greater than others, if it is possible to determine directional relation.

If is not possible to determine such a relation, return a partially evaluated result.

Assumptions are used to make the decision too.

Also, only comparable arguments are permitted.

It is named Max and not max to avoid conflicts with the built-in function max.

Examples

>>> from sympy import Max, Symbol, oo
>>> from sympy.abc import x, y
>>> p = Symbol('p', positive=True)
>>> n = Symbol('n', negative=True)

>>> Max(x, -2)                  #doctest: +SKIP
Max(x, -2)
>>> Max(x, -2).subs(x, 3)
3
>>> Max(p, -2)
p
>>> Max(x, y)
Max(x, y)
>>> Max(x, y) == Max(y, x)
True
>>> Max(x, Max(y, z))           #doctest: +SKIP
Max(x, y, z)
>>> Max(n, 8, p, 7, -oo)        #doctest: +SKIP
Max(8, p)
>>> Max (1, x, oo)
oo

Algorithm

The task can be considered as searching of supremums in the directed complete partial orders [R232].

The source values are sequentially allocated by the isolated subsets in which supremums are searched and result as Max arguments.

If the resulted supremum is single, then it is returned.

The isolated subsets are the sets of values which are only the comparable with each other in the current set. E.g. natural numbers are comparable with each other, but not comparable with the \(x\) symbol. Another example: the symbol \(x\) with negative assumption is comparable with a natural number.

Also there are “least” elements, which are comparable with all others, and have a zero property (maximum or minimum for all elements). E.g. \(oo\). In case of it the allocation operation is terminated and only this value is returned.

Assumption:

if A > B > C then A > C
if A == B then B can be removed

See also

Min: find minimum values

References

R232(1,2): https://en.wikipedia.org/wiki/Directed_complete_partial_order
R233: https://en.wikipedia.org/wiki/Lattice_%28order%29

Members

root¶

sympy.functions.elementary.miscellaneous.root(x, n, k) → Returns the k-th n-th root of x, defaulting to the[source]¶

principal root (k=0).

The parameter evaluate determines if the expression should be evaluated. If None, its value is taken from global_evaluate.

Examples

>>> from sympy import root, Rational
>>> from sympy.abc import x, n

>>> root(x, 2)
sqrt(x)

>>> root(x, 3)
x**(1/3)

>>> root(x, n)
x**(1/n)

>>> root(x, -Rational(2, 3))
x**(-3/2)

To get the k-th n-th root, specify k:

>>> root(-2, 3, 2)
-(-1)**(2/3)*2**(1/3)

To get all n n-th roots you can use the rootof function. The following examples show the roots of unity for n equal 2, 3 and 4:

>>> from sympy import rootof, I

>>> [rootof(x**2 - 1, i) for i in range(2)]
[-1, 1]

>>> [rootof(x**3 - 1,i) for i in range(3)]
[1, -1/2 - sqrt(3)*I/2, -1/2 + sqrt(3)*I/2]

>>> [rootof(x**4 - 1,i) for i in range(4)]
[-1, 1, -I, I]

SymPy, like other symbolic algebra systems, returns the complex root of negative numbers. This is the principal root and differs from the text-book result that one might be expecting. For example, the cube root of -8 does not come back as -2:

>>> root(-8, 3)
2*(-1)**(1/3)

The real_root function can be used to either make the principal result real (or simply to return the real root directly):

>>> from sympy import real_root
>>> real_root(_)
-2
>>> real_root(-32, 5)
-2

Alternatively, the n//2-th n-th root of a negative number can be computed with root:

>>> root(-32, 5, 5//2)
-2

References

sqrt¶

sympy.functions.elementary.miscellaneous.sqrt(arg, evaluate=None)[source]¶

The square root function

sqrt(x) -> Returns the principal square root of x.

The parameter evaluate determines if the expression should be evaluated. If None, its value is taken from global_evaluate

Examples

>>> from sympy import sqrt, Symbol
>>> x = Symbol('x')

>>> sqrt(x)
sqrt(x)

>>> sqrt(x)**2
x

Note that sqrt(x**2) does not simplify to x.

>>> sqrt(x**2)
sqrt(x**2)

This is because the two are not equal to each other in general. For example, consider x == -1:

>>> from sympy import Eq
>>> Eq(sqrt(x**2), x).subs(x, -1)
False

This is because sqrt computes the principal square root, so the square may put the argument in a different branch. This identity does hold if x is positive:

>>> y = Symbol('y', positive=True)
>>> sqrt(y**2)
y

You can force this simplification by using the powdenest() function with the force option set to True:

>>> from sympy import powdenest
>>> sqrt(x**2)
sqrt(x**2)
>>> powdenest(sqrt(x**2), force=True)
x

To get both branches of the square root you can use the rootof function:

>>> from sympy import rootof

>>> [rootof(x**2-3,i) for i in (0,1)]
[-sqrt(3), sqrt(3)]

References

R234: https://en.wikipedia.org/wiki/Square_root
R235: https://en.wikipedia.org/wiki/Principal_value

cbrt¶

sympy.functions.elementary.miscellaneous.cbrt(arg, evaluate=None)[source]¶

This function computes the principal cube root of \(arg\), so it’s just a shortcut for \(arg**Rational(1, 3)\).

The parameter evaluate determines if the expression should be evaluated. If None, its value is taken from global_evaluate.

Examples

>>> from sympy import cbrt, Symbol
>>> x = Symbol('x')

>>> cbrt(x)
x**(1/3)

>>> cbrt(x)**3
x

Note that cbrt(x**3) does not simplify to x.

>>> cbrt(x**3)
(x**3)**(1/3)

This is because the two are not equal to each other in general. For example, consider \(x == -1\):

>>> from sympy import Eq
>>> Eq(cbrt(x**3), x).subs(x, -1)
False

This is because cbrt computes the principal cube root, this identity does hold if \(x\) is positive:

>>> y = Symbol('y', positive=True)
>>> cbrt(y**3)
y

References

real_root¶

sympy.functions.elementary.miscellaneous.real_root(arg, n=None, evaluate=None)[source]¶

Return the real nth-root of arg if possible. If n is omitted then all instances of (-n)**(1/odd) will be changed to -n**(1/odd); this will only create a real root of a principal root – the presence of other factors may cause the result to not be real.

The parameter evaluate determines if the expression should be evaluated. If None, its value is taken from global_evaluate.

Examples

>>> from sympy import root, real_root, Rational
>>> from sympy.abc import x, n

>>> real_root(-8, 3)
-2
>>> root(-8, 3)
2*(-1)**(1/3)
>>> real_root(_)
-2

If one creates a non-principal root and applies real_root, the result will not be real (so use with caution):

>>> root(-8, 3, 2)
-2*(-1)**(2/3)
>>> real_root(_)
-2*(-1)**(2/3)