# `cmath` --- 複素数用の数学関数

このモジュールは、複素数を扱う数学関数へのアクセスを提供しています。 このモジュール中の関数は整数、浮動小数点数または複素数を引数にとります。 また、 `__complex__()` または `__float__()` どちらかのメソッドを提供している Python オブジェクトも受け付けます。 これらのメソッドはそのオブジェクトを複素数または浮動小数点数に変換するのにそれぞれ使われ、呼び出された関数はそうして変換された結果を利用します。

注釈

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

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

## 極座標変換

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`.

極座標 は複素数を表現する別の方法です。極座標では、複素数 z は半径 r と位相角 phi で定義されます。半径 r は z から原点までの距離です。位相 phi は x 軸の正の部分から原点と z を結んだ線分までの角度を反時計回りにラジアンで測った値です。

次の関数はネイティブの直交座標を極座標に変換したりその逆を行うのに使えます。

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

注釈

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))`.

## 指数関数と対数関数

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()`.

## 三角関数

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.

## 双曲線関数

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.

## 類別関数

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)

  値 a と b が互いに近い場合 `True` を、そうでない場合は `False` を返します。

  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.

  IEEE 754 特殊値 `NaN`、`inf`、`-inf` は IEEE の規則に従って処理されます。 具体的には、`NaN` は自身を含めたあらゆる値に近いとは見なされません。 `inf` と `-inf` は自身とのみ近いと見なされます。

  Added in version 3.5.

  参考

  PEP 485 -- 近似的に等しいことを調べる関数

## 定数

cmath.pi

  定数 π (円周率)で、浮動小数点数です。

cmath.e

  定数 e (自然対数の底)で、浮動小数点数です。

cmath.tau

  数学定数 τ で、浮動小数点数です。

  Added in version 3.6.

cmath.inf

  浮動小数点数の正の無限大です。`float('inf')` と等価です。

  Added in version 3.6.

cmath.infj

  実部がゼロ、虚部が正の無限大の複素数です。`complex(0.0, float('inf'))` と等価です。

  Added in version 3.6.

cmath.nan

  浮動小数点数の非数 "not a number" (NaN) です。`float('nan')` と等価です。

  Added in version 3.6.

cmath.nanj

  実部がゼロ、虚部が NaN の複素数です。`complex(0.0, float('nan'))` と等価です。

  Added in version 3.6.

`math` と同じような関数が選ばれていますが、全く同じではないので注意してください。機能を二つのモジュールに分けているのは、複素数に興味がなかったり、もしかすると複素数とは何かすら知らないようなユーザがいるからです。そういった人たちはむしろ、 `math.sqrt(-1)` が複素数を返すよりも例外を送出してほしいと考えます。また、 `cmath` で定義されている関数は、たとえ結果が実数で表現可能な場合 (虚数部がゼロの複素数) でも、常に複素数を返すので注意してください。

分枝切断 (branch cut) に関する注釈: 分枝切断を持つ曲線上では、与えられた関数は連続ではなくなります。これらは多くの複素関数における必然的な特性です。複素関数を計算する必要がある場合、これらの分枝に関して理解しているものと仮定しています。悟りに至るために何らかの (到底基礎的とはいえない) 複素数に関する書をひもといてください。数値計算を目的とした分枝切断の正しい選択方法についての情報としては、以下がよい参考文献となります:

参考

Kahan, W: Branch cuts for complex elementary functions; or, Much ado about nothings's sign bit. In Iserles, A., and Powell, M. (eds.), The state of the art in numerical analysis. Clarendon Press (1987) pp165--211.