9.3. "cmath" --- 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 "__complex__()"
or a "__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.

備註: 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.


9.3.1. Conversions to and from polar coordinates
================================================

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:

   z == z.real + z.imag*1j

*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.

cmath.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:

      >>> phase(complex(-1.0, 0.0))
      3.141592653589793
      >>> phase(complex(-1.0, -0.0))
      -3.141592653589793

備註: The modulus (absolute value) of a complex number *x* can be
  computed using the built-in "abs()" function.  There is no separate
  "cmath" module function for this operation.

cmath.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))".

cmath.rect(r, phi)

   Return the complex number *x* with polar coordinates *r* and *phi*.
   Equivalent to "r * (math.cos(phi) + math.sin(phi)*1j)".


9.3.2. Power and logarithmic functions
======================================

cmath.exp(x)

   Return the exponential value "e**x".

cmath.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.

cmath.log10(x)

   Return the base-10 logarithm of *x*. This has the same branch cut
   as "log()".

cmath.sqrt(x)

   Return the square root of *x*. This has the same branch cut as
   "log()".


9.3.3. Trigonometric functions
==============================

cmath.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.

cmath.asin(x)

   Return the arc sine of *x*. This has the same branch cuts as
   "acos()".

cmath.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.

cmath.cos(x)

   Return the cosine of *x*.

cmath.sin(x)

   Return the sine of *x*.

cmath.tan(x)

   Return the tangent of *x*.


9.3.4. Hyperbolic functions
===========================

cmath.acosh(x)

   Return the inverse hyperbolic cosine of *x*. There is one branch
   cut, extending left from 1 along the real axis to -∞, continuous
   from above.

cmath.asinh(x)

   Return the inverse hyperbolic 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.

cmath.atanh(x)

   Return the inverse hyperbolic 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.

cmath.cosh(x)

   Return the hyperbolic cosine of *x*.

cmath.sinh(x)

   Return the hyperbolic sine of *x*.

cmath.tanh(x)

   Return the hyperbolic tangent of *x*.


9.3.5. Classification functions
===============================

cmath.isfinite(x)

   Return "True" if both the real and imaginary parts of *x* are
   finite, and "False" otherwise.

   3.2 版新加入.

cmath.isinf(x)

   Return "True" if either the real or the imaginary part of *x* is an
   infinity, and "False" otherwise.

cmath.isnan(x)

   Return "True" if either the real or the imaginary part of *x* is a
   NaN, and "False" otherwise.

cmath.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.

   *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 greater than zero.

   *abs_tol* is the minimum absolute tolerance -- useful for
   comparisons near zero. *abs_tol* must be at least zero.

   If no errors occur, the result will be: "abs(a-b) <= max(rel_tol *
   max(abs(a), abs(b)), abs_tol)".

   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.

   3.5 版新加入.

   也參考: **PEP 485** -- A function for testing approximate
     equality


9.3.6. Constants
================

cmath.pi

   The mathematical constant *π*, as a float.

cmath.e

   The mathematical constant *e*, as a float.

cmath.tau

   The mathematical constant *τ*, as a float.

   3.6 版新加入.

cmath.inf

   Floating-point positive infinity. Equivalent to "float('inf')".

   3.6 版新加入.

cmath.infj

   Complex number with zero real part and positive infinity imaginary
   part. Equivalent to "complex(0.0, float('inf'))".

   3.6 版新加入.

cmath.nan

   A floating-point "not a number" (NaN) value.  Equivalent to
   "float('nan')".

   3.6 版新加入.

cmath.nanj

   Complex number with zero real part and NaN imaginary part.
   Equivalent to "complex(0.0, float('nan'))".

   3.6 版新加入.

Note that the selection of functions is similar, but not identical, to
that in module "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 "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:

也參考: 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.
