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261 lines
7.5 KiB
261 lines
7.5 KiB
: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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This module is always available. It provides access to mathematical functions
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for complex numbers. The functions in this module accept integers,
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floating-point numbers or complex numbers as arguments. They will also accept
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any Python object that has either a :meth:`__complex__` or a :meth:`__float__`
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method: these methods are used to convert the object to a complex or
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floating-point number, respectively, and the function is then applied to the
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result of the conversion.
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.. note::
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On platforms with hardware and system-level support for signed
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zeros, functions involving branch cuts are continuous on *both*
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sides of the branch cut: the sign of the zero distinguishes one
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side of the branch cut from the other. On platforms that do not
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support signed zeros the continuity is as specified below.
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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
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float. ``phase(x)`` is equivalent to ``math.atan2(x.imag,
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x.real)``. The result lies in the range [-π, π], and the branch
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cut for this operation lies along the negative real axis,
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continuous from above. On systems with support for signed zeros
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(which includes most systems in current use), this means that the
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sign of the result is the same as the sign of ``x.imag``, even when
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``x.imag`` is zero::
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>>> phase(complex(-1.0, 0.0))
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3.1415926535897931
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>>> phase(complex(-1.0, -0.0))
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-3.1415926535897931
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.. versionadded:: 2.6
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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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.. versionadded:: 2.6
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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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.. versionadded:: 2.6
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Power and logarithmic functions
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-------------------------------
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.. function:: exp(x)
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Return the exponential value ``e**x``.
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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, from 0
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along the negative real axis to -∞, continuous from above.
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.. versionchanged:: 2.4
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*base* argument added.
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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 from
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1 along the real axis to ∞, continuous from below. The other extends left from
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-1 along the real axis to -∞, continuous from above.
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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``, continuous from the right. The
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other extends from ``-1j`` along the imaginary axis to ``-∞j``, continuous
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from the left.
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.. versionchanged:: 2.6
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direction of continuity of upper cut reversed
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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 -∞, continuous from above.
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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``,
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continuous from the right. The other extends from ``-1j`` along
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the imaginary axis to ``-∞j``, continuous from the left.
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.. versionchanged:: 2.6
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branch cuts moved to match those recommended by the C99 standard
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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 ``∞``, continuous from below. The
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other extends from ``-1`` along the real axis to ``-∞``, continuous from
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above.
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.. versionchanged:: 2.6
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direction of continuity of right cut reversed
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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:: isinf(x)
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Return ``True`` if the real or the imaginary part of x is positive
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or negative infinity.
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.. versionadded:: 2.6
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.. function:: isnan(x)
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Return ``True`` if the real or imaginary part of x is not a number (NaN).
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.. versionadded:: 2.6
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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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.. index:: 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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