`cmath` — 用于复数的数学函数 ¶

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__() 或 __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.

注意

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.

例如， cmath.sqrt() function has a branch cut along the negative real axis. An argument of complex(-2.0, -0.0) is treated as though it lies below the branch cut, and so gives a result on the negative imaginary axis:

>>> cmath.sqrt(complex(-2.0, -0.0))
-1.4142135623730951j

But an argument of complex(-2.0, 0.0) is treated as though it lies above the branch cut:

>>> cmath.sqrt(complex(-2.0, 0.0))
1.4142135623730951j

转换极坐标 ¶

A Python complex number z is stored internally using rectangular or Cartesian coordinates. It is completely determined by its real part z.real 及其 imaginary part z.imag .

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) 相当于 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. The sign of the result is the same as the sign of x.imag ，甚至当 x.imag 为 0：

>>> phase(complex(-1.0, 0.0))
3.141592653589793
>>> phase(complex(-1.0, -0.0))
-3.141592653589793

注意

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 ) ¶

cmath. rect ( r , phi ) ¶

幂和对数函数 ¶

cmath. exp ( x ) ¶: 返回 e 自乘幂 x ，其中 e is the base of natural logarithms.

cmath. log ( x [ , base ] ) ¶: Returns the logarithm of x 到给定 base 。若 base is not specified, returns the natural logarithm of x . There is one branch cut, from 0 along the negative real axis to -∞.

cmath. log10 ( x ) ¶: 返回 10 基对数为 x . This has the same branch cut as log() .

cmath. sqrt ( x ) ¶: 返回平方根为 x . This has the same branch cut as log() .

三角函数 ¶

cmath. acos ( x ) ¶: 返回 acos (反余弦) 为 x . 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 ( x ) ¶: 返回 asin (反正弦) 为 x . This has the same branch cuts as acos() .

cmath. atan ( x ) ¶: 返回 atan (反正切) 为 x . 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 ( x ) ¶: 返回 cos (余弦) 为 x .

cmath. sin ( x ) ¶: 返回 sin (正弦) 为 x .

cmath. tan ( x ) ¶: 返回 tan (正切) 为 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 -∞.

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 . The other extends from -1j along the imaginary axis to -∞j .

cmath. atanh ( x ) ¶: Return the inverse hyperbolic tangent of x . 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 ( x ) ¶: Return the hyperbolic cosine of x .

cmath. sinh ( x ) ¶: Return the hyperbolic sine of x .

cmath. tanh ( x ) ¶: Return the hyperbolic tangent of x .

分类函数 ¶

cmath. isfinite ( x ) ¶

返回 True if both the real and imaginary parts of x are finite, and False 否则。

Added in version 3.2.

cmath. isinf ( x ) ¶: 返回 True if either the real or the imaginary part of x is an infinity, and False 否则。

cmath. isnan ( x ) ¶: 返回 True if either the real or the imaginary part of x is a NaN, and False 否则。

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

返回 True 若值 a and b 相互接近和 False 否则。

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) ，其是 False for any x and rel_tol less than 1.0 . So add an appropriate positive abs_tol argument to the call.

The IEEE 754 special values of NaN , inf ，和 -inf will be handled according to IEEE rules. Specifically, NaN is not considered close to any other value, including NaN . inf and -inf are only considered close to themselves.

Added in version 3.5.

另请参阅

PEP 485 – 用于测试近似相等的函数

常量 ¶

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.

Added in version 3.6.

cmath. inf ¶

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

Added in version 3.6.

cmath. infj ¶

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

Added in version 3.6.

cmath. nan ¶

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

Added in version 3.6.

cmath. nanj ¶

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

Added in version 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:

另请参阅

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.

内容表

cmath — 用于复数的数学函数
- 转换极坐标
- 幂和对数函数
  - exp()
  - log()
  - log10()
  - sqrt()
- 三角函数
  - acos()
  - asin()
  - atan()
  - cos()
  - sin()
  - tan()
- 双曲函数
  - acosh()
  - asinh()
  - atanh()
  - cosh()
  - sinh()
  - tanh()
- 分类函数
- 常量
  - pi
  - e
  - tau
  - inf
  - infj
  - nan
  - nanj

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