# Elementary¶

This module implements elementary functions, as well as functions like Abs, Max, etc.

## Abs¶

Returns the absolute value of the argument.

Examples:

>>> from sympy.functions import Abs
>>> Abs(-1)
1

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().

sign, conjugate

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.

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)


## acos¶

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

asin, atan, cos

Notes

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

Examples

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

inverse(argindex=1)[source]

Returns the inverse of this function.

## acosh¶

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

The inverse hyperbolic cosine function.

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

asinh, atanh, cosh

inverse(argindex=1)[source]

Returns the inverse of this function.

## acot¶

class sympy.functions.elementary.trigonometric.acot[source]
inverse(argindex=1)[source]

Returns the inverse of this function.

## acoth¶

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

The inverse hyperbolic cotangent function.

• acoth(x) -> Returns the inverse hyperbolic cotangent of x
inverse(argindex=1)[source]

Returns the inverse of this function.

## arg¶

Returns the argument (in radians) of a complex number. For a real 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

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

Returns the argument (in radians) of a complex number

## asin¶

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

acos, atan, sin

Notes

• asin(x) will evaluate automatically in the cases oo, -oo, 0, 1, -1

Examples

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

inverse(argindex=1)[source]

Returns the inverse of this function.

## asinh¶

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

The inverse hyperbolic sine function.

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

acosh, atanh, sinh

inverse(argindex=1)[source]

Returns the inverse of this function.

## atan¶

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

acos, asin, tan

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

inverse(argindex=1)[source]

Returns the inverse of this function.

## atan2¶

This function is like $$\operatorname{atan}$$, but considers the sign of both arguments in order to correctly determine the quadrant of its result.

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.

sin, cos, sec, csc, tan, cot, asin, acos, atan

References

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.

## atanh¶

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

The inverse hyperbolic tangent function.

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

asinh, acosh, tanh

inverse(argindex=1)[source]

Returns the inverse of this function.

## ceiling¶

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

Ceiling is a univariate function which returns the smallest integer value not less than its argument. Ceiling function is generalized in this implementation to complex numbers.

More information can be found in “Concrete mathematics” by Graham, pp. 87 or visit http://mathworld.wolfram.com/CeilingFunction.html.

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


floor

## conjugate¶

Returns the complex conjugate 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.functions import conjugate
>>> from sympy import I
>>> conjugate(2)
2
>>> conjugate(I)
-I

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

Changes the sign of the imaginary part of a complex number.

sign, Abs

Examples

>>> from sympy import conjugate, I

>>> conjugate(1 + I)
1 - I


## cos¶

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

The cosine function.

• cos(x) -> Returns the cosine of x (measured in radians)

sin, tan, acos

Notes

• cos(x) will evaluate automatically in the case x is a multiple of pi, pi/2, pi/3, pi/4 and pi/6.

References

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


## 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

sinh, tanh, acosh

## cot¶

class sympy.functions.elementary.trigonometric.cot[source]
inverse(argindex=1)[source]

Returns the inverse of this function.

## 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
inverse(argindex=1)[source]

Returns the inverse of this function.

## exp¶

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

The exponential function, $$e^x$$.

log

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

Returns this function as a 2-tuple representing a complex number.

sympy.functions.elementary.complexes.re, sympy.functions.elementary.complexes.im

Examples

>>> from sympy import I
>>> from sympy.abc import x
>>> from sympy.functions import exp
>>> exp(x).as_real_imag()
(exp(re(x))*cos(im(x)), exp(re(x))*sin(im(x)))
>>> exp(1).as_real_imag()
(E, 0)
>>> exp(I).as_real_imag()
(cos(1), sin(1))
>>> exp(1+I).as_real_imag()
(E*cos(1), E*sin(1))

base[source]

Returns the base of the exponential function.

fdiff(argindex=1)[source]

Returns the first derivative of this function.

static taylor_term(*args, **kw_args)[source]

Calculates the next term in the Taylor series expansion.

## ExprCondPair¶

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

Represents an expression, condition pair.

cond[source]

Returns the condition of this pair.

expr[source]

Returns the expression of this pair.

free_symbols[source]

Return the free symbols of this pair.

## floor¶

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

Floor is a univariate function which returns the largest integer value not greater than its argument. However this implementation generalizes floor to complex numbers.

More information can be found in “Concrete mathematics” by Graham, pp. 87 or visit http://mathworld.wolfram.com/FloorFunction.html.

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


ceiling

## HyperbolicFunction¶

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

Base class for hyperbolic functions.

sinh, cosh, tanh, coth

## IdentityFunction¶

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

The identity function

Examples

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


## im¶

Returns the imaginary part of an expression.

Examples:

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


## LambertW¶

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

Lambert W function, defined as the inverse function of x*exp(x). This function represents the principal branch of this inverse function, which like the natural logarithm is multivalued.

fdiff(argindex=1)[source]

Return the first derivative of this function.

## 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).

exp

as_base_exp()[source]

Returns this function in the form (base, exponent).

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

Returns this function as a complex coordinate.

Examples

>>> from sympy import I
>>> from sympy.abc import x
>>> from sympy.functions import log
>>> log(x).as_real_imag()
(log(Abs(x)), arg(x))
>>> log(I).as_real_imag()
(0, pi/2)
>>> log(1+I).as_real_imag()
(log(sqrt(2)), pi/4)
>>> log(I*x).as_real_imag()
(log(Abs(x)), arg(I*x))

fdiff(argindex=1)[source]

Returns the first derivative of the function.

inverse(argindex=1)[source]

Returns $$e^x$$, the inverse function of $$\log(x)$$.

static taylor_term(*args, **kw_args)[source]

Returns the next term in the Taylor series expansion of $$\log(1+x)$$.

## Min¶

Returns the minimum of two (comparable) expressions.

Examples:

>>> from sympy.functions import Min
>>> Min(1,2)
1
>>> from sympy.abc import x
>>> Min(1, x)
Min(1, x)


It is named Min and not min to avoid conflicts with the built-in function min.

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

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

Max
find maximum values

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)
Min(x, -2)

>>> Min(x, -2).subs(x, 3)
-2

>>> Min(p, -3)
-3

>>> Min(x, y)
Min(x, y)

>>> Min(n, 8, p, -7, p, oo)
Min(n, -7)


## Max¶

Returns the maximum of two (comparable) expressions

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

Min
find minimum values

References

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)
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))
Max(x, y, z)

>>> Max(n, 8, p, 7, -oo)
Max(8, p)

>>> Max (1, x, oo)
oo


Algorithm

The task can be considered as searching of supremums in the directed complete partial orders [R78].

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

## 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 a 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, a ValueError exception will be raised.
• Pairs where the cond is explicitly False, will be removed.

piecewise_fold

Examples

>>> from sympy import Piecewise, log
>>> from sympy.abc import x
>>> 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)

doit(**hints)[source]

Evaluate this piecewise function.

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

Takes an expression containing a piecewise function and returns the expression in piecewise form.

Piecewise

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, 1 <= x))


## re¶

Return the real part of an expression.

Examples:

>>> from sympy.functions import re
>>> from sympy import I
>>> re(2+3*I)
2

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.

>>> 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


im

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

Returns the real number with a zero complex part.

sympy.functions.elementary.complexes.im

## root¶

sympy.functions.elementary.miscellaneous.root(arg, n)[source]

The n-th root function (a shortcut for arg**(1/n))

root(x, n) -> Returns the principal n-th root of x.

References

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 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 (0,1) ]
[-1, 1]

>>> [ RootOf(x**3-1,i) for i in (0,1,2) ]
[1, -1/2 - sqrt(3)*I/2, -1/2 + sqrt(3)*I/2]

>>> [ RootOf(x**4-1,i) for i in (0,1,2,3) ]
[-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 such a result real or simply return the real root in the first place:

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


## RoundFunction¶

class sympy.functions.elementary.integers.RoundFunction[source]

The base class for rounding functions.

## sin¶

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

The sine function.

• sin(x) -> Returns the sine of x (measured in radians)

cos, tan, asin

Notes

• sin(x) will evaluate automatically in the case x is a multiple of pi, pi/2, pi/3, pi/4 and pi/6.

References

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


## 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

cosh, tanh, asinh

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

Returns this function as a complex coordinate.

fdiff(argindex=1)[source]

Returns the first derivative of this function.

inverse(argindex=1)[source]

Returns the inverse of this function.

static taylor_term(*args, **kw_args)[source]

Returns the next term in the Taylor series expansion.

## sqrt¶

Returns the square root of an expression. It is equivalent to raise to Rational(1,2).

>>> from sympy.functions import sqrt
>>> from sympy import Rational
>>> sqrt(2) == 2**Rational(1,2)
True

class sympy.functions.elementary.miscellaneous.sqrt[source]

The square root function

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

sympy.polys.rootoftools.RootOf, root, real_root

References

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)]


## sign¶

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

Returns the sign of an expression, that is:

• 1 if expression is positive
• 0 if expression is equal to zero
• -1 if expression is negative

Abs, conjugate

Examples

>>> from sympy.functions import sign
>>> sign(-1)
-1
>>> sign(0)
0


## tan¶

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

sin, cos, atan

Notes

• tan(x) will evaluate automatically in the case x is a multiple of pi.

References

Examples

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

inverse(argindex=1)[source]

Returns the inverse of this function.

## 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

sinh, cosh, atanh

inverse(argindex=1)[source]

Returns the inverse of this function.