:mod:`cmath` --- Mathematical functions for complex numbers

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

.. module:: cmath

   :synopsis: 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 :meth:`__complex__` or a :meth:`__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 :class:`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 :func:`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)

.. function:: phase(x)

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

   .. versionadded:: 2.6

.. function:: polar(x)

   Convert a :class:`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.

   .. versionadded:: 2.6

.. function:: rect(r, phi)

   Convert from polar coordinates to rectangular coordinates and return

   a :class:`complex`.

   .. versionadded:: 2.6

cmath functions

---------------

.. function:: 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.

.. function:: 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.

.. function:: asin(x)

   Return the arc sine of *x*. This has the same branch cuts as :func:`acos`.

.. function:: 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.

   .. versionchanged:: 2.6

      branch cuts moved to match those recommended by the C99 standard

.. function:: 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.

   .. versionchanged:: 2.6

      direction of continuity of upper cut reversed

.. function:: 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.

   .. versionchanged:: 2.6

      direction of continuity of right cut reversed

.. function:: cos(x)

   Return the cosine of *x*.

.. function:: cosh(x)

   Return the hyperbolic cosine of *x*.

.. function:: exp(x)

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

.. function:: isinf(x)

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

   or negative infinity.

   .. versionadded:: 2.6

.. function:: isnan(x)

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

   .. versionadded:: 2.6

.. function:: 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.

   .. versionchanged:: 2.4

      *base* argument added.

.. function:: log10(x)

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

   :func:`log`.

.. function:: sin(x)

   Return the sine of *x*.

.. function:: sinh(x)

   Return the hyperbolic sine of *x*.

.. function:: sqrt(x)

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

.. function:: tan(x)

   Return the tangent of *x*.

.. function:: tanh(x)

   Return the hyperbolic tangent of *x*.

The module also defines two mathematical constants:

.. data:: pi

   The mathematical constant *pi*, as a float.

.. data:: e

   The mathematical constant *e*, as a float.

.. index:: module: math

Note that the selection of functions is similar, but not identical, to that in

module :mod:`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 :mod:`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:

.. seealso::

   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.