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

This module is always available.  It 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: 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.


Complex coordinates
===================

Complex numbers can be expressed by two important coordinate systems.
Python's ``complex`` type uses rectangular coordinates where a number
on the complex plain is defined by two floats, the real part and the
imaginary part.

Definition:

   z = x + 1j * y

   x := real(z)
   y := imag(z)

In engineering the polar coordinate system is popular for complex
numbers. In polar coordinates a complex number is defined by the
radius *r* and the phase angle *phi*. The radius *r* is the absolute
value of the complex, which can be viewed as distance from (0, 0). The
radius *r* is always 0 or a positive float. The phase angle *phi* is
the counter clockwise angle from the positive x axis, e.g. *1* has the
angle *0*, *1j* has the angle *π/2* and *-1* the angle *-π*.

Note: While ``phase()`` and func:*polar* return *+π* for a negative real
  they may return *-π* for a complex with a very small negative
  imaginary part, e.g. *-1-1E-300j*.

Definition:

   z = r * exp(1j * phi)
   z = r * cis(phi)

   r := abs(z) := sqrt(real(z)**2 + imag(z)**2)
   phi := phase(z) := atan2(imag(z), real(z))
   cis(phi) := cos(phi) + 1j * sin(phi)

cmath.phase(x)

   Return phase, also known as the argument, of a complex.

cmath.polar(x)

   Convert a ``complex`` from rectangular coordinates to polar
   coordinates. The function returns a tuple with the two elements *r*
   and *phi*. *r* is the distance from 0 and *phi* the phase angle.

cmath.rect(r, phi)

   Convert from polar coordinates to rectangular coordinates and
   return a ``complex``.


cmath 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.acosh(x)

   Return the hyperbolic arc cosine of *x*. There is one branch cut,
   extending 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.asinh(x)

   Return the hyperbolic arc 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.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.atanh(x)

   Return the hyperbolic arc 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.cos(x)

   Return the cosine of *x*.

cmath.cosh(x)

   Return the hyperbolic cosine of *x*.

cmath.exp(x)

   Return the exponential value ``e**x``.

cmath.isinf(x)

   Return *True* if the real or the imaginary part of x is positive or
   negative infinity.

cmath.isnan(x)

   Return *True* if the real or imaginary part of x is not a number
   (NaN).

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.sin(x)

   Return the sine of *x*.

cmath.sinh(x)

   Return the hyperbolic sine of *x*.

cmath.sqrt(x)

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

cmath.tan(x)

   Return the tangent of *x*.

cmath.tanh(x)

   Return the hyperbolic tangent of *x*.

The module also defines two mathematical constants:

cmath.pi

   The mathematical constant *pi*, as a float.

cmath.e

   The mathematical constant *e*, as a float.

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:

See also:

   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.
