:mod:`!cmath` --- Mathematical functions for complex numbers
.. module:: cmath :synopsis: Mathematical functions for complex numbers.
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 :meth:`~object.__complex__` or a :meth:`~object.__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 :func:`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 | |
| :func:`phase(z) <phase>` | Return the phase of z |
| :func:`polar(z) <polar>` | Return the representation of z in polar coordinates |
| :func:`rect(r, phi) <rect>` | Return the complex number z with polar coordinates r and phi |
| Power and logarithmic functions | |
| :func:`exp(z) <exp>` | Return e raised to the power z |
| :func:`log(z[, base]) <log>` | Return the logarithm of z to the given base (e by default) |
| :func:`log10(z) <log10>` | Return the base-10 logarithm of z |
| :func:`sqrt(z) <sqrt>` | Return the square root of z |
| Trigonometric functions | |
| :func:`acos(z) <acos>` | Return the arc cosine of z |
| :func:`asin(z) <asin>` | Return the arc sine of z |
| :func:`atan(z) <atan>` | Return the arc tangent of z |
| :func:`cos(z) <cos>` | Return the cosine of z |
| :func:`sin(z) <sin>` | Return the sine of z |
| :func:`tan(z) <tan>` | Return the tangent of z |
| Hyperbolic functions | |
| :func:`acosh(z) <acosh>` | Return the inverse hyperbolic cosine of z |
| :func:`asinh(z) <asinh>` | Return the inverse hyperbolic sine of z |
| :func:`atanh(z) <atanh>` | Return the inverse hyperbolic tangent of z |
| :func:`cosh(z) <cosh>` | Return the hyperbolic cosine of z |
| :func:`sinh(z) <sinh>` | Return the hyperbolic sine of z |
| :func:`tanh(z) <tanh>` | Return the hyperbolic tangent of z |
| Classification functions | |
| :func:`isfinite(z) <isfinite>` | Check if all components of z are finite |
| :func:`isinf(z) <isinf>` | Check if any component of z is infinite |
| :func:`isnan(z) <isnan>` | Check if any component of z is a NaN |
| :func:`isclose(a, b, *, rel_tol, abs_tol) <isclose>` | Check if the values a and b are close to each other |
| Constants | |
| :data:`pi` | π = 3.141592... |
| :data:`e` | e = 2.718281... |
| :data:`tau` | τ = 2π = 6.283185... |
| :data:`inf` | Positive infinity |
| :data:`infj` | Pure imaginary infinity |
| :data:`nan` | "Not a number" (NaN) |
| :data:`nanj` | Pure imaginary NaN |
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.
.. function:: 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 :func:`abs` function. There is no separate :mod:`cmath` module function for this operation.
.. function:: 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))``.
.. function:: 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))``.
.. function:: exp(z) Return *e* raised to the power *z*, where *e* is the base of natural logarithms.
.. function:: log(z[, base]) Return the logarithm of *z* to the given *base*. If the *base* is not specified, returns the natural logarithm of *z*. There is one branch cut, from 0 along the negative real axis to -∞.
.. function:: log10(z) Return the base-10 logarithm of *z*. This has the same branch cut as :func:`log`.
.. function:: sqrt(z) Return the square root of *z*. This has the same branch cut as :func:`log`.
.. function:: acos(z) Return the arc cosine of *z*. There are two branch cuts: One extends right from 1 along the real axis to ∞. The other extends left from -1 along the real axis to -∞.
.. function:: asin(z) Return the arc sine of *z*. This has the same branch cuts as :func:`acos`.
.. function:: atan(z) Return the arc tangent of *z*. There are two branch cuts: One extends from ``1j`` along the imaginary axis to ``∞j``. The other extends from ``-1j`` along the imaginary axis to ``-∞j``.
.. function:: cos(z) Return the cosine of *z*.
.. function:: sin(z) Return the sine of *z*.
.. function:: tan(z) Return the tangent of *z*.
.. function:: acosh(z) Return the inverse hyperbolic cosine of *z*. There is one branch cut, extending left from 1 along the real axis to -∞.
.. function:: asinh(z) Return the inverse hyperbolic sine of *z*. There are two branch cuts: One extends from ``1j`` along the imaginary axis to ``∞j``. The other extends from ``-1j`` along the imaginary axis to ``-∞j``.
.. function:: atanh(z) Return the inverse hyperbolic tangent of *z*. There are two branch cuts: One extends from ``1`` along the real axis to ``∞``. The other extends from ``-1`` along the real axis to ``-∞``.
.. function:: cosh(z) Return the hyperbolic cosine of *z*.
.. function:: sinh(z) Return the hyperbolic sine of *z*.
.. function:: tanh(z) Return the hyperbolic tangent of *z*.
.. function:: isfinite(z) Return ``True`` if both the real and imaginary parts of *z* are finite, and ``False`` otherwise. .. versionadded:: 3.2
.. function:: isinf(z) Return ``True`` if either the real or the imaginary part of *z* is an infinity, and ``False`` otherwise.
.. function:: isnan(z) Return ``True`` if either the real or the imaginary part of *z* is a NaN, and ``False`` otherwise.
.. function:: isclose(a, b, *, rel_tol=1e-09, abs_tol=0.0)
Return ``True`` if the values *a* and *b* are close to each other and
``False`` otherwise.
Whether or not two values are considered close is determined according to
given absolute and relative tolerances. If no errors occur, the result will
be: ``abs(a-b) <= max(rel_tol * max(abs(a), abs(b)), abs_tol)``.
*rel_tol* is the relative tolerance -- it is the maximum allowed difference
between *a* and *b*, relative to the larger absolute value of *a* or *b*.
For example, to set a tolerance of 5%, pass ``rel_tol=0.05``. The default
tolerance is ``1e-09``, which assures that the two values are the same
within about 9 decimal digits. *rel_tol* must be nonnegative and less
than ``1.0``.
*abs_tol* is the absolute tolerance; it defaults to ``0.0`` and it must be
nonnegative. When comparing ``x`` to ``0.0``, ``isclose(x, 0)`` is computed
as ``abs(x) <= rel_tol * abs(x)``, which is ``False`` for any ``x`` and
rel_tol less than ``1.0``. So add an appropriate positive abs_tol argument
to the call.
The IEEE 754 special values of ``NaN``, ``inf``, and ``-inf`` will be
handled according to IEEE rules. Specifically, ``NaN`` is not considered
close to any other value, including ``NaN``. ``inf`` and ``-inf`` are only
considered close to themselves.
.. versionadded:: 3.5
.. seealso::
:pep:`485` -- A function for testing approximate equality
.. data:: pi The mathematical constant *π*, as a float.
.. data:: e The mathematical constant *e*, as a float.
.. data:: tau The mathematical constant *τ*, as a float. .. versionadded:: 3.6
.. data:: inf
Floating-point positive infinity. Equivalent to ``float('inf')``.
.. versionadded:: 3.6
.. data:: infj
Complex number with zero real part and positive infinity imaginary
part. Equivalent to ``complex(0.0, float('inf'))``.
.. versionadded:: 3.6
.. data:: nan
A floating-point "not a number" (NaN) value. Equivalent to
``float('nan')``.
.. versionadded:: 3.6
.. data:: nanj
Complex number with zero real part and NaN imaginary part. Equivalent to
``complex(0.0, float('nan'))``.
.. versionadded:: 3.6
.. index:: pair: module; math
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:
.. seealso:: 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.