Conversions to and from polar coordinates

*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.

.. function:: phase(x)

Return the phase of *x* (also known as the *argument* of *x*), as a

float. ``phase(x)`` is equivalent to ``math.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 of ``x.imag``, even when

``x.imag`` is zero::

The modulus (absolute value) of a complex number *x* can be

computed using the built-in :func:`abs` function. There is no

separate :mod:`cmath` module function for this operation.

.. function:: polar(x)

Return the representation of *x* in polar coordinates. Returns a

pair ``(r, phi)`` where *r* is the modulus of *x* and phi is the

phase of *x*. ``polar(x)`` is equivalent to ``(abs(x),

phase(x))``.

.. function:: rect(r, phi)

Return the complex number *x* with polar coordinates *r* and *phi*.

Equivalent to ``r * (math.cos(phi) + math.sin(phi)*1j)``.

Power and logarithmic functions

.. function:: exp(x)

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

.. 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:: sqrt(x)

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

Trigonometric functions

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:: asin(x)

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

.. function:: sin(x)

Return the sine of *x*.

.. function:: tan(x)

Hyperbolic functions

.. 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:: 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.

.. 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.

.. function:: cosh(x)

Return the hyperbolic cosine of *x*.

.. function:: tanh(x)

Return the hyperbolic tangent of *x*.

Classification functions

.. function:: isinf(x)

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

or negative infinity.

.. function:: isnan(x)

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

Constants

The mathematical constant *π*, as a float.