9.3. cmath — Funções matemáticas para números complexos

This module is always available. It provides access to mathematical functions for complex numbers. The functions in this module accept integers, floating-point 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 floating-point number, respectively, and the function is then applied to the result of the conversion.

Nota

On platforms with hardware and system-level 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.

9.3.1. 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 x-axis 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 to math.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 of x.imag, even when x.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 built-in 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).

9.3.2. Funções de potência e logarítmicas

cmath.exp(x)

Return the exponential value e**x.

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.

cmath.log10(x)

Return the base-10 logarithm of x. This has the same branch cut as log().

cmath.sqrt(x)

Return the square root of x. This has the same branch cut as log().

9.3.3. 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.asin(x)

Return the arc sine of x. This has the same branch cuts as acos().

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 from -1j 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.

9.3.4. 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 from -1j 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 from -1 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.

9.3.5. Classification functions

cmath.isfinite(x)

Return True if both the real and imaginary parts of x are finite, and False otherwise.

Novo na versão 3.2.

cmath.isinf(x)

Return True if either the real or the imaginary part of x is an infinity, and False otherwise.

cmath.isnan(x)

Return True if either the real or the imaginary part of x is a NaN, and False otherwise.

cmath.isclose(a, b, *, rel_tol=1e-09, abs_tol=0.0)

Retorna True se os valores a e b estiverem próximos e False 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 é 1e-09, 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(a-b) <= max(rel_tol * max(abs(a), abs(b)), abs_tol).

Os valores especiais do IEEE 754 de NaN, inf e -inf serão tratados de acordo com as regras do IEEE. Especificamente, NaN não é considerado próximo a qualquer outro valor, incluindo NaN. inf e -inf são considerados apenas próximos a si mesmos.

Novo na versão 3.5.

Ver também

PEP 485 – Uma função para testar igualdade aproximada

9.3.6. 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

Floating-point 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 floating-point “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.