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 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 Ï€/2 and -1 the angle -Ï€.
Note
While phase() and func:polar return +Ï€ for a negative real they may return -Ï€ 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)
Return phase, also known as the argument, of a complex.
New in version 2.6.
Return the hyperbolic arc sine of x. There are two branch cuts: One extends from 1j along the imaginary axis to âˆžj, continuous from the right. The other extends from -1j along the imaginary axis to -âˆžj, continuous from the left.
Changed in version 2.6: branch cuts moved to match those recommended by the C99 standard
Return the arc tangent of x. There are two branch cuts: One extends from 1j along the imaginary axis to âˆžj, continuous from the right. The other extends from -1j along the imaginary axis to -âˆžj, continuous from the left.
Changed in version 2.6: direction of continuity of upper cut reversed
Return the hyperbolic arc tangent of x. There are two branch cuts: One extends from 1 along the real axis to âˆž, continuous from below. The other extends from -1 along the real axis to -âˆž, continuous from above.
Changed in version 2.6: direction of continuity of right cut reversed
Return True if the real or the imaginary part of x is positive or negative infinity.
New in version 2.6.
Return True if the real or imaginary part of x is not a number (NaN).
New in version 2.6.
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 -âˆž, continuous from above.
Changed in version 2.4: base argument added.
The module also defines two mathematical constants:
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.