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cuts for the complex math functions. Includes a brief description of what branch cuts are.
140 lines
4.6 KiB
TeX
140 lines
4.6 KiB
TeX
\section{\module{cmath} ---
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Mathematical functions for complex numbers}
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\declaremodule{builtin}{cmath}
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\modulesynopsis{Mathematical functions for complex numbers.}
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This module is always available. It provides access to mathematical
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functions for complex numbers. The functions are:
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\begin{funcdesc}{acos}{x}
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Return the arc cosine of \var{x}.
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There are two branch cuts:
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One extends right from 1 along the real axis to \infinity, continuous
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from below.
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The other extends left from -1 along the real axis to -\infinity,
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continuous from above.
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\end{funcdesc}
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\begin{funcdesc}{acosh}{x}
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Return the hyperbolic arc cosine of \var{x}.
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There is one branch cut, extending left from 1 along the real axis
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to -\infinity, continuous from above.
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\end{funcdesc}
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\begin{funcdesc}{asin}{x}
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Return the arc sine of \var{x}.
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This has the same branch cuts as \function{acos()}.
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\end{funcdesc}
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\begin{funcdesc}{asinh}{x}
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Return the hyperbolic arc sine of \var{x}.
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There are two branch cuts, extending left from \plusminus\code{1j} to
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\plusminus-\infinity\code{j}, both continuous from above.
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These branch cuts should be considered a bug to be corrected in a
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future release.
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The correct branch cuts should extend along the imaginary axis,
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one from \code{1j} up to \infinity\code{j} and continuous from the
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right, and one from -\code{1j} down to -\infinity\code{j} and
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continuous from the left.
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\end{funcdesc}
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\begin{funcdesc}{atan}{x}
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Return the arc tangent of \var{x}.
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There are two branch cuts:
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One extends from \code{1j} along the imaginary axis to
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\infinity\code{j}, continuous from the left.
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The other extends from -\code{1j} along the imaginary axis to
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-\infinity\code{j}, continuous from the left.
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(This should probably be changed so the upper cut becomes continuous
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from the other side.)
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\end{funcdesc}
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\begin{funcdesc}{atanh}{x}
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Return the hyperbolic arc tangent of \var{x}.
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There are two branch cuts:
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One extends from 1 along the real axis to \infinity, continuous
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from above.
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The other extends from -1 along the real axis to -\infinity,
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continuous from above.
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(This should probably be changed so the right cut becomes continuous from
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the other side.)
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\end{funcdesc}
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\begin{funcdesc}{cos}{x}
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Return the cosine of \var{x}.
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\end{funcdesc}
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\begin{funcdesc}{cosh}{x}
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Return the hyperbolic cosine of \var{x}.
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\end{funcdesc}
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\begin{funcdesc}{exp}{x}
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Return the exponential value \code{e**\var{x}}.
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\end{funcdesc}
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\begin{funcdesc}{log}{x}
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Return the natural logarithm of \var{x}.
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There is one branch cut, from 0 along the negative real axis to
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-\infinity, continuous from above.
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\end{funcdesc}
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\begin{funcdesc}{log10}{x}
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Return the base-10 logarithm of \var{x}.
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This has the same branch cut as \function{log()}.
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\end{funcdesc}
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\begin{funcdesc}{sin}{x}
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Return the sine of \var{x}.
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\end{funcdesc}
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\begin{funcdesc}{sinh}{x}
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Return the hyperbolic sine of \var{x}.
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\end{funcdesc}
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\begin{funcdesc}{sqrt}{x}
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Return the square root of \var{x}.
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This has the same branch cut as \function{log()}.
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\end{funcdesc}
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\begin{funcdesc}{tan}{x}
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Return the tangent of \var{x}.
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\end{funcdesc}
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\begin{funcdesc}{tanh}{x}
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Return the hyperbolic tangent of \var{x}.
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\end{funcdesc}
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The module also defines two mathematical constants:
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\begin{datadesc}{pi}
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The mathematical constant \emph{pi}, as a real.
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\end{datadesc}
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\begin{datadesc}{e}
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The mathematical constant \emph{e}, as a real.
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\end{datadesc}
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Note that the selection of functions is similar, but not identical, to
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that in module \refmodule{math}\refbimodindex{math}. The reason for having
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two modules is that some users aren't interested in complex numbers,
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and perhaps don't even know what they are. They would rather have
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\code{math.sqrt(-1)} raise an exception than return a complex number.
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Also note that the functions defined in \module{cmath} always return a
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complex number, even if the answer can be expressed as a real number
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(in which 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
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fails to be continuous. They are a necessary feature of many complex
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functions. It is assumed that if you need to compute with complex
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functions, you will understand about branch cuts. Consult almost any
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(not too elementary) book on complex variables for enlightenment. For
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information of the proper choice of branch cuts for numerical
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purposes, a good reference should be the following:
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\begin{seealso}
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\seetext{Kahan, W: Branch cuts for complex elementary functions;
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or, Much ado about nothings's sign bit. In Iserles, A.,
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and Powell, M. (eds.), \citetitle{The state of the art in
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numerical analysis}. Clarendon Press (1987) pp165-211.}
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\end{seealso}
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