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().
See also
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.
acos¶
-
class
sympy.functions.elementary.trigonometric.
acos
[source]¶ The inverse cosine function.
Returns the arc cosine of x (measured in radians).
See also
sin
,csc
,cos
,sec
,tan
,cot
,asin
,acsc
,asec
,atan
,acot
,atan2
Notes
acos(x) will evaluate automatically in the cases oo, -oo, 0, 1, -1.
References
[R87] http://en.wikipedia.org/wiki/Inverse_trigonometric_functions [R88] http://dlmf.nist.gov/4.23 [R89] http://functions.wolfram.com/ElementaryFunctions/ArcCos Examples
>>> from sympy import acos, oo, pi >>> acos(1) 0 >>> acos(0) pi/2 >>> acos(oo) oo*I
acosh¶
acot¶
-
class
sympy.functions.elementary.trigonometric.
acot
[source]¶ The inverse cotangent function.
Returns the arc cotangent of x (measured in radians).
See also
sin
,csc
,cos
,sec
,tan
,cot
,asin
,acsc
,acos
,asec
,atan
,atan2
References
[R90] http://en.wikipedia.org/wiki/Inverse_trigonometric_functions [R91] http://dlmf.nist.gov/4.23 [R92] http://functions.wolfram.com/ElementaryFunctions/ArcCot
acoth¶
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
asin¶
-
class
sympy.functions.elementary.trigonometric.
asin
[source]¶ The inverse sine function.
Returns the arcsine of x in radians.
See also
sin
,csc
,cos
,sec
,tan
,cot
,acsc
,acos
,asec
,atan
,acot
,atan2
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).
References
[R93] http://en.wikipedia.org/wiki/Inverse_trigonometric_functions [R94] http://dlmf.nist.gov/4.23 [R95] http://functions.wolfram.com/ElementaryFunctions/ArcSin Examples
>>> from sympy import asin, oo, pi >>> asin(1) pi/2 >>> asin(-1) -pi/2
asinh¶
atan¶
-
class
sympy.functions.elementary.trigonometric.
atan
[source]¶ The inverse tangent function.
Returns the arc tangent of x (measured in radians).
See also
sin
,csc
,cos
,sec
,tan
,cot
,asin
,acsc
,acos
,asec
,acot
,atan2
Notes
atan(x) will evaluate automatically in the cases oo, -oo, 0, 1, -1.
References
[R96] http://en.wikipedia.org/wiki/Inverse_trigonometric_functions [R97] http://dlmf.nist.gov/4.23 [R98] http://functions.wolfram.com/ElementaryFunctions/ArcTan Examples
>>> from sympy import atan, oo, pi >>> atan(0) 0 >>> atan(1) pi/4 >>> atan(oo) pi/2
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:\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}
Attention: Note the role reversal of both arguments. The y-coordinate is the first argument and the x-coordinate the second.
See also
sin
,csc
,cos
,sec
,tan
,cot
,asin
,acsc
,acos
,asec
,atan
,acot
References
[R99] http://en.wikipedia.org/wiki/Inverse_trigonometric_functions [R100] http://en.wikipedia.org/wiki/Atan2 [R101] http://functions.wolfram.com/ElementaryFunctions/ArcTan2 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¶
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
See also
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
cos¶
-
class
sympy.functions.elementary.trigonometric.
cos
[source]¶ The cosine function.
Returns the cosine of x (measured in radians).
See also
sin
,csc
,sec
,tan
,cot
,asin
,acsc
,acos
,asec
,atan
,acot
,atan2
Notes
See
sin()
for notes about automatic evaluation.References
[R102] http://en.wikipedia.org/wiki/Trigonometric_functions [R103] http://dlmf.nist.gov/4.14 [R104] http://functions.wolfram.com/ElementaryFunctions/Cos 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¶
cot¶
-
class
sympy.functions.elementary.trigonometric.
cot
[source]¶ The cotangent function.
Returns the cotangent of x (measured in radians).
See also
sin
,csc
,cos
,sec
,tan
,asin
,acsc
,acos
,asec
,atan
,acot
,atan2
Notes
See
sin()
for notes about automatic evaluation.References
[R105] http://en.wikipedia.org/wiki/Trigonometric_functions [R106] http://dlmf.nist.gov/4.14 [R107] http://functions.wolfram.com/ElementaryFunctions/Cot Examples
>>> from sympy import cot >>> from sympy.abc import x >>> cot(x**2).diff(x) 2*x*(-cot(x**2)**2 - 1) >>> cot(1).diff(x) 0
coth¶
exp¶
-
class
sympy.functions.elementary.exponential.
exp
[source]¶ The exponential function, e^x.
See also
log
-
as_real_imag
(deep=True, **hints)[source]¶ Returns this function as a 2-tuple representing a complex number.
See also
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
¶ Returns the base of the exponential function.
-
See also
ExprCondPair¶
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
See also
ceiling
HyperbolicFunction¶
IdentityFunction¶
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]¶ The Lambert W function W(z) is defined as the inverse function of w \exp(w) [R108].
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.
References
[R108] (1, 2) http://en.wikipedia.org/wiki/Lambert_W_function 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
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
, uselog(x, b)
, which is essentially short-hand forlog(x)/log(b)
.See also
exp
-
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))
-
See also
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.
See also
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.
See also
Min
- find minimum values
References
[R109] (1, 2) http://en.wikipedia.org/wiki/Directed_complete_partial_order [R110] http://en.wikipedia.org/wiki/Lattice_%28order%29 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 [R109].
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 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, a ValueError exception will be raised.
- Pairs where the cond is explicitly False, will be removed.
See also
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)
-
sympy.functions.elementary.piecewise.
piecewise_fold
(expr)[source]¶ Takes an expression containing a piecewise function and returns the expression in piecewise form.
See also
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
See also
im
See also
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.
See also
sympy.polys.rootoftools.RootOf
,sympy.core.power.integer_nthroot
,sqrt
,real_root
References
- http://en.wikipedia.org/wiki/Square_root
- http://en.wikipedia.org/wiki/Real_root
- http://en.wikipedia.org/wiki/Root_of_unity
- http://en.wikipedia.org/wiki/Principal_value
- http://mathworld.wolfram.com/CubeRoot.html
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¶
sin¶
-
class
sympy.functions.elementary.trigonometric.
sin
[source]¶ The sine function.
Returns the sine of x (measured in radians).
See also
csc
,cos
,sec
,tan
,cot
,asin
,acsc
,acos
,asec
,atan
,acot
,atan2
Notes
This function will evaluate automatically in the case x/pi is some rational number [R114]. For example, if x is a multiple of pi, pi/2, pi/3, pi/4 and pi/6.
References
[R111] http://en.wikipedia.org/wiki/Trigonometric_functions [R112] http://dlmf.nist.gov/4.14 [R113] http://functions.wolfram.com/ElementaryFunctions/Sin [R114] (1, 2) http://mathworld.wolfram.com/TrigonometryAngles.html 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¶
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.
See also
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 complex sign of an expression:
If the expresssion 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 expresssion 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))
.See also
Abs
,conjugate
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
tan¶
-
class
sympy.functions.elementary.trigonometric.
tan
[source]¶ The tangent function.
Returns the tangent of x (measured in radians).
See also
sin
,csc
,cos
,sec
,cot
,asin
,acsc
,acos
,asec
,atan
,acot
,atan2
Notes
See
sin()
for notes about automatic evaluation.References
[R115] http://en.wikipedia.org/wiki/Trigonometric_functions [R116] http://dlmf.nist.gov/4.14 [R117] http://functions.wolfram.com/ElementaryFunctions/Tan 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