"cmath" --- Matematyczne funkcje dla liczb zespolonych
******************************************************

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

Ten moduł zapewnia dostęp do funkcji matematycznych dla liczb
zespolonych. Funkcje w tym module akceptują liczby całkowite, liczby
zmiennoprzecinkowe lub liczby zespolone jako argumenty. Funkcje w tym
module zaakceptują również obiekty Python, które zawierają metody
"__complex__()" lub "__float__()": te metody są używane do
konwertowania obiektów do liczb zmiennoprzecinkowych lub zespolonych,
odpowiednio, a funkcje używają później wyników konwersji.

Informacja:

  On platforms with hardware and system-level support for signed
  zeros, functions involving branch cuts are continuous on *both*
  sides of the branch cut: the sign of the zero distinguishes one side
  of the branch cut from the other.  On platforms that do not support
  signed zeros the continuity is as specified below.


Conversions to and from polar coordinates
=========================================

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".  In other words:

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

*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)" 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, continuous from
   above.  On systems with support for signed zeros (which includes
   most systems in current use), this means that 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

Informacja:

  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)

   Return the representation of *x* in polar coordinates.  Returns a
   pair "(r, phi)" where *r* is the modulus of *x* and phi is the
   phase of *x*.  "polar(x)" is equivalent to "(abs(x), phase(x))".

cmath.rect(r, phi)

   Return the complex number *x* with polar coordinates *r* and *phi*.
   Equivalent to "r * (math.cos(phi) + math.sin(phi)*1j)".


Power and logarithmic functions
===============================

cmath.exp(x)

   Return *e* raised to the power *x*, where *e* is the base of
   natural logarithms.

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 -∞, continuous
   from above.

cmath.log10(x)

   Return the base-10 logarithm of *x*. This has the same branch cut
   as "log()".

cmath.sqrt(x)

   Return the square root of *x*. This has the same branch cut as
   "log()".


Trigonometric functions
=======================

cmath.acos(x)

   Return the arc cosine of *x*. There are two branch cuts: One
   extends right from 1 along the real axis to ∞, continuous from
   below. The other extends left from -1 along the real axis to -∞,
   continuous from above.

cmath.asin(x)

   Return the arc sine of *x*. This has the same branch cuts as
   "acos()".

cmath.atan(x)

   Return the arc tangent of *x*. There are two branch cuts: One
   extends from "1j" along the imaginary axis to "∞j", continuous from
   the right. The other extends from "-1j" along the imaginary axis to
   "-∞j", continuous from the left.

cmath.cos(x)

   Return the cosine of *x*.

cmath.sin(x)

   Return the sine of *x*.

cmath.tan(x)

   Return the tangent of *x*.


Hyperbolic functions
====================

cmath.acosh(x)

   Return the inverse hyperbolic cosine of *x*. There is one branch
   cut, extending left from 1 along the real axis to -∞, continuous
   from above.

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",
   continuous from the right.  The other extends from "-1j" along the
   imaginary axis to "-∞j", continuous from the left.

cmath.atanh(x)

   Return the inverse hyperbolic tangent of *x*. There are two branch
   cuts: One extends from "1" along the real axis to "∞", continuous
   from below. The other extends from "-1" along the real axis to
   "-∞", continuous from above.

cmath.cosh(x)

   Zwraca cosinus hiperboliczny z  *x*.

cmath.sinh(x)

   Zwraca sinus hiperboliczny z *x*.

cmath.tanh(x)

   Zwraca tangens hiperboliczny z *x*.


Funkcje klasyfikujące
=====================

cmath.isfinite(x)

   Zwraca 1 "True" jeżeli obie rzeczywista i urojona część *x* są
   skończone, i 2``False`` w innym wypadku.

   Nowe w wersji 3.2.

cmath.isinf(x)

   Return "True" if either the real or the imaginary part of *x* is an
   infinity, and "False" otherwise.

cmath.isnan(x)

   Return "True" if either the real or the imaginary part of *x* is a
   NaN, and "False" otherwise.

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

   Zwraca 1``True`` jeżeli wartości *a* i *b* są zbliżone do siebie i
   2``False`` w innym wypadku.

   To czy dwie wartości są zbliżone do siebie, zależy od dostarczonej
   absolutnej lub relatywnej tolerancji.

   *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 greater than zero.

   *abs_tol* is the minimum absolute tolerance -- useful for
   comparisons near zero. *abs_tol* must be at least zero.

   If no errors occur, the result will be: "abs(a-b) <= max(rel_tol *
   max(abs(a), abs(b)), abs_tol)".

   The IEEE 754 special values of "NaN", "inf", and "-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.

   Nowe w wersji 3.5.

   Zobacz także:

     **PEP 485** -- A function for testing approximate equality


Stały
=====

cmath.pi

   Stałą matematyczną *π*, jako liczbę zmiennoprzecinkową

cmath.e

   Stałą matematyczną *e*, jako liczbę zmiennoprzecinkową

cmath.tau

   Stałą matematyczną *τ*, jako liczbę zmiennoprzecinkową

   Nowe w wersji 3.6.

cmath.inf

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

   Nowe w wersji 3.6.

cmath.infj

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

   Nowe w wersji 3.6.

cmath.nan

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

   Nowe w wersji 3.6.

cmath.nanj

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

   Nowe w wersji 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:

Zobacz także:

  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.
