"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 "-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            |
+------------------------------------------------------+--------------------------------------------------------------------+
| **定数**                                                                                                                  |
+------------------------------------------------------+--------------------------------------------------------------------+
| "pi"                                                 | *π* = 3.141592...                                                  |
+------------------------------------------------------+--------------------------------------------------------------------+
| "e"                                                  | *e* = 2.718281...                                                  |
+------------------------------------------------------+--------------------------------------------------------------------+
| "tau"                                                | *τ* = 2*π* = 6.283185...                                           |
+------------------------------------------------------+--------------------------------------------------------------------+
| "inf"                                                | 正の無限大                                                         |
+------------------------------------------------------+--------------------------------------------------------------------+
| "infj"                                               | Pure imaginary infinity                                            |
+------------------------------------------------------+--------------------------------------------------------------------+
| "nan"                                                | 非数 "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(-1+0j)
      3.141592653589793
      >>> phase(-1-0j)
      -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

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

   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.

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).

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.
