"cmath" --- Mathematical functions for complex numbers
******************************************************

======================================================================

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__()" 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.

Note:

  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* = 2*π* = 6.283185...                                           |
+------------------------------------------------------+--------------------------------------------------------------------+
| "inf"                                                | Positive infinity                                                  |
+------------------------------------------------------+--------------------------------------------------------------------+
| "infj"                                               | Pure imaginary infinity                                            |
+------------------------------------------------------+--------------------------------------------------------------------+
| "nan"                                                | "Not a number" (NaN)                                               |
+------------------------------------------------------+--------------------------------------------------------------------+
| "nanj"                                               | Pure imaginary NaN                                                 |
+------------------------------------------------------+--------------------------------------------------------------------+


Conversion vers et à partir de coordonnées polaires
===================================================

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".

Les *coordonnées polaires* donnent une manière alternative de
représenter un nombre complexe. En coordonnées polaires, un nombre
complexe *z* est défini par son module *r* et par son argument (*angle
de phase*) *phi*. Le module *r* est la distance entre *z* et
l'origine, alors que l'argument *phi* est l'angle (dans le sens
inverse des aiguilles d'une montre, ou sens trigonométrique), mesuré
en radians, à partir de l'axe X positif, et vers le segment de droite
joignant *z* à l'origine.

Les fonctions suivantes peuvent être utilisées pour convertir à partir
des coordonnées rectangulaires natives vers les coordonnées polaires,
et vice-versa.

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

Note:

  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))".


Fonctions logarithme et exponentielle
=====================================

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()".


Fonctions trigonométriques
==========================

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*.


Fonctions hyperboliques
=======================

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*.


Fonctions de classifications
============================

cmath.isfinite(z)

   Return "True" if both the real and imaginary parts of *z* are
   finite, and "False" otherwise.

   Ajouté dans la 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)

   Renvoie "True" si les valeurs *a* et *b* sont proches l'une de
   l'autre, et "False" sinon.

   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.

   Les valeurs spécifiques suivantes : "NaN", "inf", et "-inf"
   définies dans la norme IEEE 754  seront manipulées selon les règles
   du standard IEEE. En particulier, "NaN" n'est considéré proche
   d'aucune autre valeur, "NaN" inclus. "inf" et "-inf" ne sont
   considérés proches que d'eux-mêmes.

   Ajouté dans la version 3.5.

   Voir aussi:

     **PEP 485** -- Une fonction pour tester des égalités approximées


Constantes
==========

cmath.pi

   La constante mathématique *π*, en tant que flottant.

cmath.e

   La constante mathématique *e*, en tant que flottant.

cmath.tau

   La constante mathématique *τ*, sous forme de flottant.

   Ajouté dans la version 3.6.

cmath.inf

   Nombre à virgule flottante positif infini. Équivaut à
   "float('inf')".

   Ajouté dans la version 3.6.

cmath.infj

   Nombre complexe dont la partie réelle vaut zéro et la partie
   imaginaire un infini positif. Équivalent à "complex(0.0,
   float('inf'))".

   Ajouté dans la version 3.6.

cmath.nan

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

   Ajouté dans la version 3.6.

cmath.nanj

   Nombre complexe dont la partie réelle vaut zéro et la partie
   imaginaire vaut un *NaN*. Équivalent à "complex(0.0,
   float('nan'))".

   Ajouté dans la version 3.6.

Notez que la sélection de fonctions est similaire, mais pas identique,
à celles du module "math". La raison d'avoir deux modules est que
certains utilisateurs ne sont pas intéressés par les nombres
complexes, et peut-être ne savent même pas ce qu'ils sont. Ils
préféreraient alors que "math.sqrt(-1)" lève une exception au lieu de
renvoyer un nombre complexe. Également, notez que les fonctions
définies dans "cmath" renvoient toujours un nombre complexe, même si
le résultat peut être exprimé à l'aide d'un nombre réel (en quel cas
la partie imaginaire du complexe vaut zéro).

Une note sur les *coupures* : ce sont des courbes sur lesquelles la
fonction n'est pas continue. Ce sont des caractéristiques nécessaires
de beaucoup de fonctions complexes. Il est supposé que si vous avez
besoin d'utiliser des fonctions complexes, vous comprendrez ce que
sont les coupures. Consultez n'importe quel livre (pas trop
élémentaire) sur les variables complexes pour plus d'informations.
Pour des informations sur les choix des coupures à des fins
numériques, voici une bonne référence :

Voir aussi:

  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.
