# `cmath` — Mathematical functions for complex numbers

This module 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

For functions involving branch cuts, we have the problem of deciding how to define those functions on the cut itself. Following Kahan’s «Branch cuts for complex elementary functions» paper, as well as Annex G of C99 and later C standards, we use the sign of zero to distinguish one side of the branch cut from the other: for a branch cut along (a portion of) the real axis we look at the sign of the imaginary part, while for a branch cut along the imaginary axis we look at the sign of the real part.

For example, the `cmath.sqrt()` function has a branch cut along the negative real axis. An argument of `-2-0j` is treated as though it lies below the branch cut, and so gives a result on the negative imaginary axis:

    >>> cmath.sqrt(-2-0j)
    -1.4142135623730951j

But an argument of `-2+0j` is treated as though it lies above the branch cut:

    >>> cmath.sqrt(-2+0j)
    1.4142135623730951j

Conversions to and from polar coordinates
`phase(z)` | Return the phase of z
`polar(z)` | Return the representation of z in polar coordinates
`rect(r, phi)` | Return the complex number z with polar coordinates r and phi
Power and logarithmic functions
`exp(z)` | Return e raised to the power z
`log(z[, base])` | Return the logarithm of z to the given base (e by default)
`log10(z)` | Return the base-10 logarithm of z
`sqrt(z)` | Return the square root of z
Trigonometric functions
`acos(z)` | Return the arc cosine of z
`asin(z)` | Return the arc sine of z
`atan(z)` | Return the arc tangent of z
`cos(z)` | Return the cosine of z
`sin(z)` | Return the sine of z
`tan(z)` | Return the tangent of z
Hyperbolic functions
`acosh(z)` | Return the inverse hyperbolic cosine of z
`asinh(z)` | Return the inverse hyperbolic sine of z
`atanh(z)` | Return the inverse hyperbolic tangent of z
`cosh(z)` | Return the hyperbolic cosine of z
`sinh(z)` | Return the hyperbolic sine of z
`tanh(z)` | Return the hyperbolic tangent of z
Classification functions
`isfinite(z)` | Check if all components of z are finite
`isinf(z)` | Check if any component of z is infinite
`isnan(z)` | Check if any component of z is a NaN
`isclose(a, b, *, rel_tol, abs_tol)` | Check if the values a and b are close to each other
Constants
`pi` | π = 3.141592…
`e` | e = 2.718281…
`tau` | τ = 2π = 6.283185…
`inf` | Positive infinity
`infj` | Pure imaginary infinity
`nan` | «Not a number» (NaN)
`nanj` | Pure imaginary NaN

## Conversión a y desde coordenadas polares

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`.

Las coordenadas polares dan una alternativa a la representación de números complejos. En las coordenadas polares, un número complejo z se define por los módulos r y el ángulo de fase phi. El módulo r es la distancia desde z hasta el origen, mientras que la fase phi es el ángulo que va en contra de las agujas del reloj, medido en radianes, desde el eje positivo de las X hasta el segmento de linea que une el origen con z.

Las siguientes funciones pueden ser usadas para convertir desde coordenadas rectangulares nativas hasta coordenadas polares y viceversa.

cmath.phase(z)

  Return the phase of z (also known as the argument of z), as a float. `phase(z)` is equivalent to `math.atan2(z.imag, z.real)`. The result lies in the range [-π, π], and the branch cut for this operation lies along the negative real axis. The sign of the result is the same as the sign of `z.imag`, even when `z.imag` is zero:

      >>> phase(-1+0j)
      3.141592653589793
      >>> phase(-1-0j)
      -3.141592653589793

Nota

The modulus (absolute value) of a complex number z can be computed using the built-in `abs()` function. There is no separate `cmath` module function for this operation.

cmath.polar(z)

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

cmath.rect(r, phi)

  Return the complex number z with polar coordinates r and phi. Equivalent to `complex(r * math.cos(phi), r * math.sin(phi))`.

## Funciones logarítmicas y de potencias

cmath.exp(z)

  Return e raised to the power z, where e is the base of natural logarithms.

cmath.log(z[, base])

  Return the logarithm of z to the given base. If the base is not specified, returns the natural logarithm of z. There is one branch cut, from 0 along the negative real axis to -∞.

cmath.log10(z)

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

cmath.sqrt(z)

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

## Funciones trigonométricas

cmath.acos(z)

  Return the arc cosine of z. There are two branch cuts: One extends right from 1 along the real axis to ∞. The other extends left from -1 along the real axis to -∞.

cmath.asin(z)

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

cmath.atan(z)

  Return the arc tangent of z. There are two branch cuts: One extends from `1j` along the imaginary axis to `∞j`. The other extends from `-1j` along the imaginary axis to `-∞j`.

cmath.cos(z)

  Return the cosine of z.

cmath.sin(z)

  Return the sine of z.

cmath.tan(z)

  Return the tangent of z.

## Funciones hiperbólicas

cmath.acosh(z)

  Return the inverse hyperbolic cosine of z. There is one branch cut, extending left from 1 along the real axis to -∞.

cmath.asinh(z)

  Return the inverse hyperbolic sine of z. There are two branch cuts: One extends from `1j` along the imaginary axis to `∞j`. The other extends from `-1j` along the imaginary axis to `-∞j`.

cmath.atanh(z)

  Return the inverse hyperbolic tangent of z. There are two branch cuts: One extends from `1` along the real axis to `∞`. The other extends from `-1` along the real axis to `-∞`.

cmath.cosh(z)

  Return the hyperbolic cosine of z.

cmath.sinh(z)

  Return the hyperbolic sine of z.

cmath.tanh(z)

  Return the hyperbolic tangent of z.

## Funciones de clasificación

cmath.isfinite(z)

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

  Added in version 3.2.

cmath.isinf(z)

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

cmath.isnan(z)

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

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

  Retorna `True` si los valores a y b son cercanos el uno al otro y `Falso` de otro modo.

  Whether or not two values are considered close is determined according to given absolute and relative tolerances. If no errors occur, the result will be: `abs(a-b) <= max(rel_tol * max(abs(a), abs(b)), abs_tol)`.

  rel_tol is the relative tolerance – it is the maximum allowed difference between a and b, relative to the larger absolute value of a or b. For example, to set a tolerance of 5%, pass `rel_tol=0.05`. The default tolerance is `1e-09`, which assures that the two values are the same within about 9 decimal digits. rel_tol must be nonnegative and less than `1.0`.

  abs_tol is the absolute tolerance; it defaults to `0.0` and it must be nonnegative. When comparing `x` to `0.0`, `isclose(x, 0)` is computed as `abs(x) <= rel_tol * abs(x)`, which is `False` for any `x` and rel_tol less than `1.0`. So add an appropriate positive abs_tol argument to the call.

  Los valores especiales IEEE 754 de `NaN`, `inf` y `-inf` serán manejados de acuerdo al estándar de IEEE. Especialmente, `NaN` no se considera cercano a ningún otro valor, incluido `NaN`. `inf` y `-inf` solo son considerados cercanos a sí mismos.

  Added in version 3.5.

  Ver también

  PEP 485 – Una función para probar igualdad aproximada.

## Constantes

cmath.pi

  La constante matemática π, como número de coma flotante.

cmath.e

  La constante matemática e, como número de coma flotante.

cmath.tau

  La constante matemática τ, como número de coma flotante.

  Added in version 3.6.

cmath.inf

  Números de coma flotante de +infinito. Equivalente a `float('inf')`.

  Added in version 3.6.

cmath.infj

  Números complejos con la parte real cero y números positivos infinitos como la parte imaginaria. Equivalente a `complex(0.0, float('inf'))`.

  Added in version 3.6.

cmath.nan

  A floating-point «not a number» (NaN) value. Equivalent to `float('nan')`. See also `math.nan`.

  Added in version 3.6.

cmath.nanj

  Números complejos con parte real cero y como parte imaginaria NaN. Equivalente a `complex(0.0, float('nan'))`.

  Added in version 3.6.

Nótese que la selección de funciones es similar, pero no idéntica, a la del módulo `math`. El motivo de tener dos módulos se halla en que algunos usuarios no se encuentran interesados en números complejos, y quizás ni siquiera sepan que son. Preferirían que `math.sqrt(-1)` lance una excepción a que retorne un número complejo. Además fíjese que las funciones definidas en `cmath` siempre retornan un número complejo, incluso si la respuesta puede ser expresada como un número real (en cuyo caso el número complejo tiene una parte imaginaria de cero).

Un apunte en los tramos: Se tratan de curvas en las cuales las funciones fallan a ser continua. Son un complemento necesario de muchas funciones complejas. Se asume que si se necesitan cálculos con funciones complejas, usted entenderá sobre tramos. Consulte casi cualquier(no muy elemental) libro sobre variables complejas para saber más. Para más información en la correcta elección de los tramos para propósitos numéricos, se recomienda la siguiente bibliografía:

Ver también

Kahan, W: Branch cuts for complex elementary functions; o, Much ado about nothing’s sign bit. En Iserles, A., and Powell, M. (eds.), The state of the art in numerical analysis. Clarendon Press (1987) pp165–211.