mirror of
https://github.com/python/cpython.git
synced 2024-12-04 15:25:13 +08:00
228 lines
6.5 KiB
ReStructuredText
228 lines
6.5 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 *phi*. 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 *phi* 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 * phi)
|
|
z = r * cis(phi)
|
|
|
|
r := abs(z) := sqrt(real(z)**2 + imag(z)**2)
|
|
phi := phase(z) := atan2(imag(z), real(z))
|
|
cis(phi) := cos(phi) + 1j * sin(phi)
|
|
|
|
|
|
.. function:: phase(x)
|
|
|
|
Return phase, also known as the argument, of a complex.
|
|
|
|
|
|
.. 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.
|
|
|
|
|
|
.. function:: rect(r, phi)
|
|
|
|
Convert from polar coordinates to rectangular coordinates and return
|
|
a :class:`complex`.
|
|
|
|
|
|
|
|
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.
|
|
|
|
|
|
.. 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.
|
|
|
|
|
|
.. 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:: 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.
|
|
|
|
|
|
.. function:: isnan(x)
|
|
|
|
Return *True* if the real or imaginary part of x is not a number (NaN).
|
|
|
|
|
|
.. 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.
|
|
|
|
|