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.

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 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 theargumentofx), as a float.phase(x)is equivalent tomath.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 ofx.imag, even whenx.imagis zero:>>> phase(complex(-1.0, 0.0)) 3.1415926535897931 >>> phase(complex(-1.0, -0.0)) -3.1415926535897931New in version 2.6.

- Note: 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

xin polar coordinates. Returns a pair(r, phi)whereris the modulus ofxand phi is the phase ofx.polar(x)is equivalent to(abs(x), phase(x)).New in version 2.6.

cmath.rect(r, phi)

Return the complex number

xwith polar coordinatesrandphi. Equivalent tor * (math.cos(phi) + math.sin(phi)*1j).New in version 2.6.

cmath.exp(x)

Return the exponential valuee**x.

cmath.log(x[, base])

Returns the logarithm of

xto the givenbase. If thebaseis not specified, returns the natural logarithm ofx. There is one branch cut, from 0 along the negative real axis to -∞, continuous from above.Changed in version 2.4:

baseargument added.

cmath.log10(x)

Return the base-10 logarithm ofx. This has the same branch cut aslog().

cmath.sqrt(x)

Return the square root ofx. This has the same branch cut aslog().

cmath.acos(x)

Return the arc cosine ofx. 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 ofx. This has the same branch cuts asacos().

cmath.atan(x)

Return the arc tangent of

x. There are two branch cuts: One extends from1jalong the imaginary axis to∞j, continuous from the right. The other extends from-1jalong the imaginary axis to-∞j, continuous from the left.Changed in version 2.6: direction of continuity of upper cut reversed

cmath.cos(x)

Return the cosine ofx.

cmath.sin(x)

Return the sine ofx.

cmath.tan(x)

Return the tangent ofx.

cmath.acosh(x)

Return the hyperbolic arc cosine ofx. There is one branch cut, extending left from 1 along the real axis to -∞, continuous from above.

cmath.asinh(x)

Return the hyperbolic arc sine of

x. There are two branch cuts: One extends from1jalong the imaginary axis to∞j, continuous from the right. The other extends from-1jalong the imaginary axis to-∞j, continuous from the left.Changed in version 2.6: branch cuts moved to match those recommended by the C99 standard

cmath.atanh(x)

Return the hyperbolic arc tangent of

x. There are two branch cuts: One extends from1along the real axis to∞, continuous from below. The other extends from-1along the real axis to-∞, continuous from above.Changed in version 2.6: direction of continuity of right cut reversed

cmath.cosh(x)

Return the hyperbolic cosine ofx.

cmath.sinh(x)

Return the hyperbolic sine ofx.

cmath.tanh(x)

Return the hyperbolic tangent ofx.

cmath.isinf(x)

Return

Trueif the real or the imaginary part of x is positive or negative infinity.New in version 2.6.

cmath.isnan(x)

Return

Trueif the real or imaginary part of x is not a number (NaN).New in version 2.6.

cmath.pi

The mathematical constantπ, as a float.

cmath.e

The mathematical constante, 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.