Mirror
/
cpython
mirror of https://github.com/python/cpython
4
1
Fork 0
Code Issues Releases Wiki Activity

Merged revisions 74184,74230 via svnmerge from 

svn+ssh://pythondev@svn.python.org/python/trunk

........
  r74184 | georg.brandl | 2009-07-23 08:08:58 +0100 (Thu, 23 Jul 2009) | 1 line

  #6548: dont suggest existence of real and imag functions in cmath.
........
  r74230 | mark.dickinson | 2009-07-28 17:12:40 +0100 (Tue, 28 Jul 2009) | 4 lines

  Issue #6458: Reorganize cmath documentation into sections (similar to
  the way that the math documentation is organized); clarify section on
  conversions to and from polar coordinates.
........
This commit is contained in:
Mark Dickinson 2009-07-28 16:31:03 +00:00
parent e805ecc752
commit c2eab8908d
1 changed files with 116 additions and 102 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

218
Doc/library/cmath.rst
View File

@ -22,66 +22,95 @@ result of the conversion.
support signed zeros the continuity is as specified below.
Complex coordinates
-------------------
Conversions to and from polar 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.
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``. In other
words::
Definition::
z == z.real + z.imag*1j
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)
*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 phase, also known as the argument, of a complex.
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::
>>> phase(complex(-1.0, 0.0))
3.141592653589793
>>> phase(complex(-1.0, -0.0))
-3.141592653589793
.. note::
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)
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.
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)
Convert from polar coordinates to rectangular coordinates and return
a :class:`complex`.
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
-------------------------------
cmath 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
-----------------------
.. function:: acos(x)
@ -90,25 +119,11 @@ cmath functions
-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
@ -117,6 +132,38 @@ cmath functions
from the left.
.. function:: cos(x)
Return the cosine of *x*.
.. function:: sin(x)
Return the sine of *x*.
.. function:: tan(x)
Return the tangent of *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
@ -125,21 +172,24 @@ cmath functions
above.
.. function:: cos(x)
Return the cosine of *x*.
.. function:: cosh(x)
Return the hyperbolic cosine of *x*.
.. function:: exp(x)
.. function:: sinh(x)
Return the exponential value ``e**x``.
Return the hyperbolic sine 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
@ -151,49 +201,13 @@ cmath functions
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:
Constants
---------
.. data:: pi
The mathematical constant *pi*, as a float.
The mathematical constant *π*, as a float.
.. data:: e