mirror of https://github.com/python/cpython
327 lines
9.4 KiB
ReStructuredText
327 lines
9.4 KiB
ReStructuredText
:mod:`!cmath` --- Mathematical functions for complex numbers
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============================================================
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.. module:: cmath
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:synopsis: Mathematical functions for complex numbers.
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--------------
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This module provides access to mathematical functions for complex numbers. The
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functions in this module accept integers, floating-point numbers or complex
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numbers as arguments. They will also accept any Python object that has either a
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:meth:`~object.__complex__` or a :meth:`~object.__float__` method: these methods are used to
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convert the object to a complex or floating-point number, respectively, and
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the function is then applied to the result of the conversion.
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.. note::
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For functions involving branch cuts, we have the problem of deciding how to
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define those functions on the cut itself. Following Kahan's "Branch cuts for
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complex elementary functions" paper, as well as Annex G of C99 and later C
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standards, we use the sign of zero to distinguish one side of the branch cut
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from the other: for a branch cut along (a portion of) the real axis we look
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at the sign of the imaginary part, while for a branch cut along the
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imaginary axis we look at the sign of the real part.
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For example, the :func:`cmath.sqrt` function has a branch cut along the
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negative real axis. An argument of ``complex(-2.0, -0.0)`` is treated as
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though it lies *below* the branch cut, and so gives a result on the negative
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imaginary axis::
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>>> cmath.sqrt(complex(-2.0, -0.0))
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-1.4142135623730951j
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But an argument of ``complex(-2.0, 0.0)`` is treated as though it lies above
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the branch cut::
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>>> cmath.sqrt(complex(-2.0, 0.0))
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1.4142135623730951j
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Conversions to and from polar coordinates
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-----------------------------------------
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A Python complex number ``z`` is stored internally using *rectangular*
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or *Cartesian* coordinates. It is completely determined by its *real
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part* ``z.real`` and its *imaginary part* ``z.imag``. In other
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words::
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z == z.real + z.imag*1j
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*Polar coordinates* give an alternative way to represent a complex
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number. In polar coordinates, a complex number *z* is defined by the
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modulus *r* and the phase angle *phi*. The modulus *r* is the distance
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from *z* to the origin, while the phase *phi* is the counterclockwise
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angle, measured in radians, from the positive x-axis to the line
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segment that joins the origin to *z*.
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The following functions can be used to convert from the native
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rectangular coordinates to polar coordinates and back.
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.. function:: phase(x)
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Return the phase of *x* (also known as the *argument* of *x*), as a float.
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``phase(x)`` is equivalent to ``math.atan2(x.imag, x.real)``. The result
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lies in the range [-\ *π*, *π*], and the branch cut for this operation lies
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along the negative real axis. The sign of the result is the same as the
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sign of ``x.imag``, even when ``x.imag`` is zero::
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>>> phase(complex(-1.0, 0.0))
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3.141592653589793
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>>> phase(complex(-1.0, -0.0))
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-3.141592653589793
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.. note::
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The modulus (absolute value) of a complex number *x* can be
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computed using the built-in :func:`abs` function. There is no
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separate :mod:`cmath` module function for this operation.
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.. function:: polar(x)
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Return the representation of *x* in polar coordinates. Returns a
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pair ``(r, phi)`` where *r* is the modulus of *x* and phi is the
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phase of *x*. ``polar(x)`` is equivalent to ``(abs(x),
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phase(x))``.
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.. function:: rect(r, phi)
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Return the complex number *x* with polar coordinates *r* and *phi*.
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Equivalent to ``r * (math.cos(phi) + math.sin(phi)*1j)``.
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Power and logarithmic functions
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-------------------------------
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.. function:: exp(x)
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Return *e* raised to the power *x*, where *e* is the base of natural
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logarithms.
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.. function:: log(x[, base])
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Returns the logarithm of *x* to the given *base*. If the *base* is not
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specified, returns the natural logarithm of *x*. There is one branch cut,
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from 0 along the negative real axis to -∞.
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.. function:: log10(x)
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Return the base-10 logarithm of *x*. This has the same branch cut as
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:func:`log`.
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.. function:: sqrt(x)
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Return the square root of *x*. This has the same branch cut as :func:`log`.
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Trigonometric functions
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-----------------------
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.. function:: acos(x)
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Return the arc cosine of *x*. There are two branch cuts: One extends right
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from 1 along the real axis to ∞. The other extends left from -1 along the
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real axis to -∞.
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.. function:: asin(x)
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Return the arc sine of *x*. This has the same branch cuts as :func:`acos`.
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.. function:: atan(x)
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Return the arc tangent of *x*. There are two branch cuts: One extends from
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``1j`` along the imaginary axis to ``∞j``. The other extends from ``-1j``
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along the imaginary axis to ``-∞j``.
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.. function:: cos(x)
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Return the cosine of *x*.
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.. function:: sin(x)
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Return the sine of *x*.
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.. function:: tan(x)
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Return the tangent of *x*.
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Hyperbolic functions
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--------------------
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.. function:: acosh(x)
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Return the inverse hyperbolic cosine of *x*. There is one branch cut,
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extending left from 1 along the real axis to -∞.
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.. function:: asinh(x)
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Return the inverse hyperbolic sine of *x*. There are two branch cuts:
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One extends from ``1j`` along the imaginary axis to ``∞j``. The other
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extends from ``-1j`` along the imaginary axis to ``-∞j``.
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.. function:: atanh(x)
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Return the inverse hyperbolic tangent of *x*. There are two branch cuts: One
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extends from ``1`` along the real axis to ``∞``. The other extends from
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``-1`` along the real axis to ``-∞``.
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.. function:: cosh(x)
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Return the hyperbolic cosine of *x*.
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.. function:: sinh(x)
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Return the hyperbolic sine of *x*.
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.. function:: tanh(x)
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Return the hyperbolic tangent of *x*.
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Classification functions
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------------------------
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.. function:: isfinite(x)
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Return ``True`` if both the real and imaginary parts of *x* are finite, and
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``False`` otherwise.
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.. versionadded:: 3.2
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.. function:: isinf(x)
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Return ``True`` if either the real or the imaginary part of *x* is an
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infinity, and ``False`` otherwise.
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.. function:: isnan(x)
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Return ``True`` if either the real or the imaginary part of *x* is a NaN,
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and ``False`` otherwise.
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.. function:: isclose(a, b, *, rel_tol=1e-09, abs_tol=0.0)
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Return ``True`` if the values *a* and *b* are close to each other and
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``False`` otherwise.
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Whether or not two values are considered close is determined according to
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given absolute and relative tolerances.
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*rel_tol* is the relative tolerance -- it is the maximum allowed difference
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between *a* and *b*, relative to the larger absolute value of *a* or *b*.
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For example, to set a tolerance of 5%, pass ``rel_tol=0.05``. The default
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tolerance is ``1e-09``, which assures that the two values are the same
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within about 9 decimal digits. *rel_tol* must be greater than zero.
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*abs_tol* is the minimum absolute tolerance -- useful for comparisons near
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zero. *abs_tol* must be at least zero.
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If no errors occur, the result will be:
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``abs(a-b) <= max(rel_tol * max(abs(a), abs(b)), abs_tol)``.
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The IEEE 754 special values of ``NaN``, ``inf``, and ``-inf`` will be
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handled according to IEEE rules. Specifically, ``NaN`` is not considered
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close to any other value, including ``NaN``. ``inf`` and ``-inf`` are only
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considered close to themselves.
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.. versionadded:: 3.5
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.. seealso::
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:pep:`485` -- A function for testing approximate equality
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Constants
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---------
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.. data:: pi
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The mathematical constant *π*, as a float.
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.. data:: e
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The mathematical constant *e*, as a float.
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.. data:: tau
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The mathematical constant *τ*, as a float.
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.. versionadded:: 3.6
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.. data:: inf
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Floating-point positive infinity. Equivalent to ``float('inf')``.
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.. versionadded:: 3.6
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.. data:: infj
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Complex number with zero real part and positive infinity imaginary
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part. Equivalent to ``complex(0.0, float('inf'))``.
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.. versionadded:: 3.6
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.. data:: nan
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A floating-point "not a number" (NaN) value. Equivalent to
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``float('nan')``.
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.. versionadded:: 3.6
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.. data:: nanj
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Complex number with zero real part and NaN imaginary part. Equivalent to
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``complex(0.0, float('nan'))``.
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.. versionadded:: 3.6
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.. index:: pair: module; math
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Note that the selection of functions is similar, but not identical, to that in
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module :mod:`math`. The reason for having two modules is that some users aren't
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interested in complex numbers, and perhaps don't even know what they are. They
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would rather have ``math.sqrt(-1)`` raise an exception than return a complex
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number. Also note that the functions defined in :mod:`cmath` always return a
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complex number, even if the answer can be expressed as a real number (in which
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case the complex number has an imaginary part of zero).
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A note on branch cuts: They are curves along which the given function fails to
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be continuous. They are a necessary feature of many complex functions. It is
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assumed that if you need to compute with complex functions, you will understand
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about branch cuts. Consult almost any (not too elementary) book on complex
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variables for enlightenment. For information of the proper choice of branch
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cuts for numerical purposes, a good reference should be the following:
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.. seealso::
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Kahan, W: Branch cuts for complex elementary functions; or, Much ado about
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nothing's sign bit. In Iserles, A., and Powell, M. (eds.), The state of the art
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in numerical analysis. Clarendon Press (1987) pp165--211.
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