# `cmath` --- Mathematical functions for complex numbers

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.

注解

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.

## 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

注解

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)`.

## 幂函数与对数函数

`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.

`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()`.

## 三角函数

`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)

  Return the sine of x.

`cmath.``tan`(x)

  Return the tangent of x.

## 双曲函数

`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)

  返回 x 的双曲余弦值。

`cmath.``sinh`(x)

  返回 x 的双曲正弦值。

`cmath.``tanh`(x)

  返回 x 的双曲正切值。

## Classification functions

`cmath.``isfinite`(x)

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

  3.2 新版功能.

`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)

  若 a 和 b 的值比较接近则返回 `True`，否则返回 `False`。

  根据给定的绝对和相对容差确定两个值是否被认为是接近的。

  rel_tol 是相对容差 —— 它是 a 和 b 之间允许的最大差值，相对于 a 或 b 的较大绝对值。例如，要设置5％的容差，请传递 `rel_tol=0.05` 。默认容差为 `1e-09`，确保两个值在大约9位十进制数字内相同。 rel_tol 必须大于零。

  abs_tol 是最小绝对容差 —— 对于接近零的比较很有用。 abs_tol 必须至少为零。

  如果没有错误发生，结果将是： `abs(a-b) <= max(rel_tol * max(abs(a), abs(b)), abs_tol)` 。

  IEEE 754特殊值 `NaN` ， `inf` 和` -inf` 将根据IEEE规则处理。具体来说， `NaN` 不被认为接近任何其他值，包括 `NaN` 。 `inf` 和 `-inf` 只被认为接近自己。

  3.5 新版功能.

  参见

  PEP 485 —— 用于测试近似相等的函数

## 常数

`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.

  3.6 新版功能.

`cmath.``inf`

  Floating-point positive infinity. Equivalent to `float('inf')`.

  3.6 新版功能.

`cmath.``infj`

  Complex number with zero real part and positive infinity imaginary part. Equivalent to `complex(0.0, float('inf'))`.

  3.6 新版功能.

`cmath.``nan`

  A floating-point "not a number" (NaN) value. Equivalent to `float('nan')`.

  3.6 新版功能.

`cmath.``nanj`

  Complex number with zero real part and NaN imaginary part. Equivalent to `complex(0.0, float('nan'))`.

  3.6 新版功能.

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:

参见

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.