mirror of
https://github.com/python/cpython.git
synced 2025-07-24 11:44:31 +00:00
53876d9cd8
svn+ssh://pythondev@svn.python.org/python/trunk ........ r62380 | christian.heimes | 2008-04-19 01:13:07 +0200 (Sat, 19 Apr 2008) | 3 lines I finally got the time to update and merge Mark's and my trunk-math branch. The patch is collaborated work of Mark Dickinson and me. It was mostly done a few months ago. The patch fixes a lot of loose ends and edge cases related to operations with NaN, INF, very small values and complex math. The patch also adds acosh, asinh, atanh, log1p and copysign to all platforms. Finally it fixes differences between platforms like different results or exceptions for edge cases. Have fun :) ........ r62382 | christian.heimes | 2008-04-19 01:40:40 +0200 (Sat, 19 Apr 2008) | 2 lines Added new files to Windows project files More Windows related fixes are coming soon ........ r62383 | christian.heimes | 2008-04-19 01:49:11 +0200 (Sat, 19 Apr 2008) | 1 line Stupid me. Py_RETURN_NAN should actually return something ... ........
247 lines
6.8 KiB
ReStructuredText
247 lines
6.8 KiB
ReStructuredText
|
|
:mod:`cmath` --- Mathematical functions for complex numbers
|
|
===========================================================
|
|
|
|
.. module:: cmath
|
|
:synopsis: Mathematical functions for complex numbers.
|
|
|
|
|
|
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 :meth:`__complex__` or a :meth:`__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 coordinates
|
|
-------------------
|
|
|
|
Complex numbers can be expressed by two important coordinate systems.
|
|
Python's :class:`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 *φ*. 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 *φ* is the counter clockwise angle from the positive x axis,
|
|
e.g. *1* has the angle *0*, *1j* has the angle *π/2* and *-1* the angle *-π*.
|
|
|
|
.. note::
|
|
While :func:`phase` and func:`polar` return *+π* for a negative real they
|
|
may return *-π* for a complex with a very small negative imaginary
|
|
part, e.g. *-1-1E-300j*.
|
|
|
|
|
|
Definition::
|
|
|
|
z = r * exp(1j * φ)
|
|
z = r * cis(φ)
|
|
|
|
r := abs(z) := sqrt(real(z)**2 + imag(z)**2)
|
|
phi := phase(z) := atan2(imag(z), real(z))
|
|
cis(φ) := cos(φ) + 1j * sin(φ)
|
|
|
|
|
|
.. function:: phase(x)
|
|
|
|
Return phase, also known as the argument, of a complex.
|
|
|
|
.. versionadded:: 2.6
|
|
|
|
|
|
.. function:: polar(x)
|
|
|
|
Convert a :class:`complex` from rectangular coordinates to polar
|
|
coordinates. The function returns a tuple with the two elements
|
|
*r* and *phi*. *r* is the distance from 0 and *phi* the phase
|
|
angle.
|
|
|
|
.. versionadded:: 2.6
|
|
|
|
|
|
.. function:: rect(r, phi)
|
|
|
|
Convert from polar coordinates to rectangular coordinates and return
|
|
a :class:`complex`.
|
|
|
|
.. versionadded:: 2.6
|
|
|
|
|
|
|
|
cmath functions
|
|
---------------
|
|
|
|
.. function:: acos(x)
|
|
|
|
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:: 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:: asin(x)
|
|
|
|
Return the arc sine of *x*. This has the same branch cuts as :func:`acos`.
|
|
|
|
|
|
.. 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.
|
|
|
|
.. versionchanged:: 2.6
|
|
branch cuts moved to match those recommended by the C99 standard
|
|
|
|
|
|
.. function:: atan(x)
|
|
|
|
Return the arc tangent 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.
|
|
|
|
.. versionchanged:: 2.6
|
|
direction of continuity of upper cut reversed
|
|
|
|
|
|
.. 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.
|
|
|
|
.. versionchanged:: 2.6
|
|
direction of continuity of right cut reversed
|
|
|
|
|
|
.. function:: cos(x)
|
|
|
|
Return the cosine of *x*.
|
|
|
|
|
|
.. function:: cosh(x)
|
|
|
|
Return the hyperbolic cosine of *x*.
|
|
|
|
|
|
.. function:: exp(x)
|
|
|
|
Return the exponential value ``e**x``.
|
|
|
|
|
|
.. function:: isinf(x)
|
|
|
|
Return *True* if the real or the imaginary part of x is positive
|
|
or negative infinity.
|
|
|
|
.. versionadded:: 2.6
|
|
|
|
|
|
.. function:: isnan(x)
|
|
|
|
Return *True* if the real or imaginary part of x is not a number (NaN).
|
|
|
|
.. versionadded:: 2.6
|
|
|
|
|
|
.. 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.
|
|
|
|
|
|
.. function:: log10(x)
|
|
|
|
Return the base-10 logarithm of *x*. This has the same branch cut as
|
|
:func:`log`.
|
|
|
|
|
|
.. function:: sin(x)
|
|
|
|
Return the sine of *x*.
|
|
|
|
|
|
.. function:: sinh(x)
|
|
|
|
Return the hyperbolic sine of *x*.
|
|
|
|
|
|
.. function:: sqrt(x)
|
|
|
|
Return the square root of *x*. This has the same branch cut as :func:`log`.
|
|
|
|
|
|
.. function:: tan(x)
|
|
|
|
Return the tangent of *x*.
|
|
|
|
|
|
.. function:: tanh(x)
|
|
|
|
Return the hyperbolic tangent of *x*.
|
|
|
|
The module also defines two mathematical constants:
|
|
|
|
|
|
.. data:: pi
|
|
|
|
The mathematical constant *pi*, as a float.
|
|
|
|
|
|
.. data:: e
|
|
|
|
The mathematical constant *e*, as a float.
|
|
|
|
.. index:: 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.
|
|
|
|
|