cmath
— Mathematical functions for complex numbers¶
This module provides access to mathematical functions for complex numbers. The
functions in this module accept integers, floatingpoint numbers or complex
numbers as arguments. They will also accept any Python object that has either a
__complex__()
or a __float__()
method: these methods are used to
convert the object to a complex or floatingpoint number, respectively, and
the function is then applied to the result of the conversion.
Nota
On platforms with hardware and systemlevel support for signed zeros, functions involving branch cuts are continuous on both sides of the branch cut: the sign of the zero distinguishes one side of the branch cut from the other. On platforms that do not support signed zeros the continuity is as specified below.
Conversions to and from polar coordinates¶
A Python complex number z
is stored internally using rectangular
or Cartesian coordinates. It is completely determined by its real
part z.real
and its imaginary part z.imag
. In other
words:
z == z.real + z.imag*1j
Polar coordinates give an alternative way to represent a complex number. In polar coordinates, a complex number z is defined by the modulus r and the phase angle phi. The modulus r is the distance from z to the origin, while the phase phi is the counterclockwise angle, measured in radians, from the positive xaxis to the line segment that joins the origin to z.
The following functions can be used to convert from the native rectangular coordinates to polar coordinates and back.

cmath.
phase
(x)¶ Return the phase of x (also known as the argument of x), as a float.
phase(x)
is equivalent tomath.atan2(x.imag, x.real)
. The result lies in the range [π, π], and the branch cut for this operation lies along the negative real axis, continuous from above. On systems with support for signed zeros (which includes most systems in current use), this means that the sign of the result is the same as the sign ofx.imag
, even whenx.imag
is zero:>>> phase(complex(1.0, 0.0)) 3.141592653589793 >>> phase(complex(1.0, 0.0)) 3.141592653589793
Nota
The modulus (absolute value) of a complex number x can be
computed using the builtin abs()
function. There is no
separate cmath
module function for this operation.

cmath.
polar
(x)¶ Return the representation of x in polar coordinates. Returns a pair
(r, phi)
where r is the modulus of x and phi is the phase of x.polar(x)
is equivalent to(abs(x), phase(x))
.

cmath.
rect
(r, phi)¶ Return the complex number x with polar coordinates r and phi. Equivalent to
r * (math.cos(phi) + math.sin(phi)*1j)
.
Funções de potência e logarítmicas¶

cmath.
exp
(x)¶ Return e raised to the power x, where e is the base of natural logarithms.

cmath.
log
(x[, base])¶ Returns the logarithm of x to the given base. If the base is not specified, returns the natural logarithm of x. There is one branch cut, from 0 along the negative real axis to ∞, continuous from above.
Funções trigonométricas¶

cmath.
acos
(x)¶ Return the arc cosine of x. There are two branch cuts: One extends right from 1 along the real axis to ∞, continuous from below. The other extends left from 1 along the real axis to ∞, continuous from above.

cmath.
atan
(x)¶ Return the arc tangent of x. There are two branch cuts: One extends from
1j
along the imaginary axis to∞j
, continuous from the right. The other extends from1j
along the imaginary axis to∞j
, continuous from the left.

cmath.
cos
(x)¶ Return the cosine of x.

cmath.
sin
(x)¶ Devolve o seno de x.

cmath.
tan
(x)¶ Return the tangent of x.
Funções hiperbólicas¶

cmath.
acosh
(x)¶ Return the inverse hyperbolic cosine of x. There is one branch cut, extending left from 1 along the real axis to ∞, continuous from above.

cmath.
asinh
(x)¶ Return the inverse hyperbolic sine of x. There are two branch cuts: One extends from
1j
along the imaginary axis to∞j
, continuous from the right. The other extends from1j
along the imaginary axis to∞j
, continuous from the left.

cmath.
atanh
(x)¶ Return the inverse hyperbolic tangent of x. There are two branch cuts: One extends from
1
along the real axis to∞
, continuous from below. The other extends from1
along the real axis to∞
, continuous from above.

cmath.
cosh
(x)¶ Retorna o cosseno hiperbólico de x.

cmath.
sinh
(x)¶ Retorna o seno hiperbólico de x.

cmath.
tanh
(x)¶ Retorna a tangente hiperbólica de x.
Classification functions¶

cmath.
isfinite
(x)¶ Return
True
if both the real and imaginary parts of x are finite, andFalse
otherwise.Novo na versão 3.2.

cmath.
isinf
(x)¶ Return
True
if either the real or the imaginary part of x is an infinity, andFalse
otherwise.

cmath.
isnan
(x)¶ Return
True
if either the real or the imaginary part of x is a NaN, andFalse
otherwise.

cmath.
isclose
(a, b, *, rel_tol=1e09, abs_tol=0.0)¶ Retorna
True
se os valores a e b estiverem próximos eFalse
caso contrário.Se dois valores são ou não considerados próximos, é determinado de acordo com as tolerâncias absolutas e relativas fornecidas.
rel_tol é a tolerância relativa – é a diferença máxima permitida entre a e b, em relação ao maior valor absoluto de a e b. Por exemplo, para definir uma tolerância de 5%, passe
rel_tol=0.05
. A tolerância padrão é1e09
, o que garante que os dois valores sejam iguais em cerca de 9 dígitos decimais. rel_tol deve ser maior que zero.abs_tol é a tolerância absoluta mínima – útil para comparações próximas a zero. abs_tol deve ser pelo menos zero.
Se nenhum erro ocorrer, o resultado será:
abs(ab) <= max(rel_tol * max(abs(a), abs(b)), abs_tol)
.Os valores especiais do IEEE 754 de
NaN
,inf
einf
serão tratados de acordo com as regras do IEEE. Especificamente,NaN
não é considerado próximo a qualquer outro valor, incluindoNaN
.inf
einf
são considerados apenas próximos a si mesmos.Novo na versão 3.5.
Ver também
PEP 485 – Uma função para testar igualdade aproximada
Constantes¶

cmath.
pi
¶ The mathematical constant π, as a float.

cmath.
e
¶ The mathematical constant e, as a float.

cmath.
tau
¶ The mathematical constant τ, as a float.
Novo na versão 3.6.

cmath.
inf
¶ Floatingpoint positive infinity. Equivalent to
float('inf')
.Novo na versão 3.6.

cmath.
infj
¶ Complex number with zero real part and positive infinity imaginary part. Equivalent to
complex(0.0, float('inf'))
.Novo na versão 3.6.

cmath.
nan
¶ A floatingpoint “not a number” (NaN) value. Equivalent to
float('nan')
.Novo na versão 3.6.

cmath.
nanj
¶ Complex number with zero real part and NaN imaginary part. Equivalent to
complex(0.0, float('nan'))
.Novo na versão 3.6.
Note that the selection of functions is similar, but not identical, to that in
module math
. The reason for having two modules is that some users aren’t
interested in complex numbers, and perhaps don’t even know what they are. They
would rather have math.sqrt(1)
raise an exception than return a complex
number. Also note that the functions defined in cmath
always return a
complex number, even if the answer can be expressed as a real number (in which
case the complex number has an imaginary part of zero).
A note on branch cuts: They are curves along which the given function fails to be continuous. They are a necessary feature of many complex functions. It is assumed that if you need to compute with complex functions, you will understand about branch cuts. Consult almost any (not too elementary) book on complex variables for enlightenment. For information of the proper choice of branch cuts for numerical purposes, a good reference should be the following:
Ver também
Kahan, W: Branch cuts for complex elementary functions; or, Much ado about nothing’s sign bit. In Iserles, A., and Powell, M. (eds.), The state of the art in numerical analysis. Clarendon Press (1987) pp165–211.