cmath — Mathematical functions for complex numbers

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This module 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

For functions involving branch cuts, we have the problem of deciding how to define those functions on the cut itself. Following Kahan’s “Branch cuts for complex elementary functions” paper, as well as Annex G of C99 and later C standards, we use the sign of zero to distinguish one side of the branch cut from the other: for a branch cut along (a portion of) the real axis we look at the sign of the imaginary part, while for a branch cut along the imaginary axis we look at the sign of the real part.

For example, the cmath.sqrt() function has a branch cut along the negative real axis. An argument of -2-0j is treated as though it lies below the branch cut, and so gives a result on the negative imaginary axis:

>>> cmath.sqrt(-2-0j)
-1.4142135623730951j

But an argument of -2+0j is treated as though it lies above the branch cut:

>>> cmath.sqrt(-2+0j)
1.4142135623730951j

Conversions to and from polar coordinates
phase(z)	Return the phase of z
polar(z)	Return the representation of z in polar coordinates
rect(r, phi)	Return the complex number z with polar coordinates r and phi
Power and logarithmic functions
exp(z)	Return e raised to the power z
log(z[, base])	Return the logarithm of z to the given base (e by default)
log10(z)	Return the base-10 logarithm of z
sqrt(z)	Return the square root of z
Trigonometric functions
acos(z)	Return the arc cosine of z
asin(z)	Return the arc sine of z
atan(z)	Return the arc tangent of z
cos(z)	Return the cosine of z
sin(z)	Return the sine of z
tan(z)	Return the tangent of z
Hyperbolic functions
acosh(z)	Return the inverse hyperbolic cosine of z
asinh(z)	Return the inverse hyperbolic sine of z
atanh(z)	Return the inverse hyperbolic tangent of z
cosh(z)	Return the hyperbolic cosine of z
sinh(z)	Return the hyperbolic sine of z
tanh(z)	Return the hyperbolic tangent of z
Classification functions
isfinite(z)	Check if all components of z are finite
isinf(z)	Check if any component of z is infinite
isnan(z)	Check if any component of z is a NaN
isclose(a, b, *, rel_tol, abs_tol)	Check if the values a and b are close to each other
Constants
pi	π = 3.141592…
e	e = 2.718281…
tau	τ = 2π = 6.283185…
inf	Positive infinity
infj	Pure imaginary infinity
nan	“Not a number” (NaN)
nanj	Pure imaginary NaN

Conversions to and from polar coordinates

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.

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, measured in radians, 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(z)

Return the phase of z (also known as the argument of z), as a float. phase(z) is equivalent to math.atan2(z.imag, z.real). The result lies in the range [-π, π], and the branch cut for this operation lies along the negative real axis. The sign of the result is the same as the sign of z.imag, even when z.imag is zero:

>>> phase(-1+0j)
3.141592653589793
>>> phase(-1-0j)
-3.141592653589793

Note

The modulus (absolute value) of a complex number z can be computed using the built-in abs() function. There is no separate cmath module function for this operation.

cmath.polar(z)

Return the representation of z in polar coordinates. Returns a pair (r, phi) where r is the modulus of z and phi is the phase of z. polar(z) is equivalent to (abs(z), phase(z)).

cmath.rect(r, phi)

Return the complex number z with polar coordinates r and phi. Equivalent to complex(r * math.cos(phi), r * math.sin(phi)).