`cmath` --- 关于复数的数学函数¶

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 `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
```

到极坐标和从极坐标的转换¶

```z == z.real + z.imag*1j
```

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. 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
```

cmath.polar(x)

cmath.rect(r, phi)

幂函数与对数函数¶

cmath.exp(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 -∞.

cmath.log10(x)

cmath.sqrt(x)

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 ∞. The other extends left from -1 along the real axis to -∞.

cmath.asin(x)

cmath.atan(x)

Return the arc tangent 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.cos(x)

cmath.sin(x)

cmath.tan(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)

cmath.sinh(x)

cmath.tanh(x)

分类函数¶

cmath.isfinite(x)

3.2 新版功能.

cmath.isinf(x)

cmath.isnan(x)

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

ab 的值比较接近则返回 `True`，否则返回 `False`

rel_tol 是相对容差 —— 它是 ab 之间允许的最大差值，相对于 ab 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。

IEEE 754特殊值 `NaN``inf``-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``-inf` 只被认为接近自己。

3.5 新版功能.

PEP 485 —— 用于测试近似相等的函数

常量¶

cmath.pi

cmath.e

cmath.tau

3.6 新版功能.

cmath.inf

3.6 新版功能.

cmath.infj

3.6 新版功能.

cmath.nan

3.6 新版功能.

cmath.nanj

3.6 新版功能.

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.