cmath.rst



:mod:`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 :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)


cmath functions
The module also defines two mathematical constants:
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: