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228 lines
6.4 KiB
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: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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Complex coordinates
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-------------------
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Complex numbers can be expressed by two important coordinate systems.
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Python's :class:`complex` type uses rectangular coordinates where a number
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on the complex plain is defined by two floats, the real part and the imaginary
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part.
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Definition::
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z = x + 1j * y
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x := real(z)
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y := imag(z)
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In engineering the polar coordinate system is popular for complex numbers. In
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polar coordinates a complex number is defined by the radius *r* and the phase
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angle *φ*. The radius *r* is the absolute value of the complex, which can be
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viewed as distance from (0, 0). The radius *r* is always 0 or a positive float.
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The phase angle *φ* is the counter clockwise angle from the positive x axis,
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e.g. *1* has the angle *0*, *1j* has the angle *π/2* and *-1* the angle *-π*.
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.. note::
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While :func:`phase` and func:`polar` return *+π* for a negative real they
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may return *-π* for a complex with a very small negative imaginary
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part, e.g. *-1-1E-300j*.
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Definition::
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z = r * exp(1j * φ)
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z = r * cis(φ)
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r := abs(z) := sqrt(real(z)**2 + imag(z)**2)
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phi := phase(z) := atan2(imag(z), real(z))
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cis(φ) := cos(φ) + 1j * sin(φ)
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.. function:: phase(x)
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Return phase, also known as the argument, of a complex.
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.. function:: polar(x)
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Convert a :class:`complex` from rectangular coordinates to polar
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coordinates. The function returns a tuple with the two elements
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*r* and *phi*. *r* is the distance from 0 and *phi* the phase
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angle.
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.. function:: rect(r, phi)
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Convert from polar coordinates to rectangular coordinates and return
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a :class:`complex`.
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cmath 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:: acosh(x)
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Return the hyperbolic arc cosine of *x*. There is one branch cut, extending left
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from 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:: asinh(x)
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Return the hyperbolic arc 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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.. 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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.. function:: atanh(x)
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Return the hyperbolic arc 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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.. function:: cos(x)
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Return the cosine of *x*.
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.. function:: cosh(x)
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Return the hyperbolic cosine of *x*.
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.. function:: exp(x)
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Return the exponential value ``e**x``.
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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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.. 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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.. 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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.. 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:: sin(x)
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Return the sine of *x*.
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.. function:: sinh(x)
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Return the hyperbolic sine of *x*.
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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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.. function:: tan(x)
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Return the tangent of *x*.
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.. function:: tanh(x)
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Return the hyperbolic tangent of *x*.
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The module also defines two mathematical constants:
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.. data:: pi
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The mathematical constant *pi*, 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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