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diff --git a/Doc/lib/libcmath.tex b/Doc/lib/libcmath.tex index 3a11870..77496bf 100644 --- a/Doc/lib/libcmath.tex +++ b/Doc/lib/libcmath.tex @@ -9,26 +9,56 @@ functions for complex numbers. The functions are: \begin{funcdesc}{acos}{x} Return the arc cosine of \var{x}. +There are two branch cuts: +One extends right from 1 along the real axis to \infinity, continuous +from below. +The other extends left from -1 along the real axis to -\infinity, +continuous from above. \end{funcdesc} \begin{funcdesc}{acosh}{x} Return the hyperbolic arc cosine of \var{x}. +There is one branch cut, extending left from 1 along the real axis +to -\infinity, continuous from above. \end{funcdesc} \begin{funcdesc}{asin}{x} Return the arc sine of \var{x}. +This has the same branch cuts as \function{acos()}. \end{funcdesc} \begin{funcdesc}{asinh}{x} Return the hyperbolic arc sine of \var{x}. +There are two branch cuts, extending left from \plusminus\code{1j} to +\plusminus-\infinity\code{j}, both continuous from above. +These branch cuts should be considered a bug to be corrected in a +future release. +The correct branch cuts should extend along the imaginary axis, +one from \code{1j} up to \infinity\code{j} and continuous from the +right, and one from -\code{1j} down to -\infinity\code{j} and +continuous from the left. \end{funcdesc} \begin{funcdesc}{atan}{x} Return the arc tangent of \var{x}. +There are two branch cuts: +One extends from \code{1j} along the imaginary axis to +\infinity\code{j}, continuous from the left. +The other extends from -\code{1j} along the imaginary axis to +-\infinity\code{j}, continuous from the left. +(This should probably be changed so the upper cut becomes continuous +from the other side.) \end{funcdesc} \begin{funcdesc}{atanh}{x} Return the hyperbolic arc tangent of \var{x}. +There are two branch cuts: +One extends from 1 along the real axis to \infinity, continuous +from above. +The other extends from -1 along the real axis to -\infinity, +continuous from above. +(This should probably be changed so the right cut becomes continuous from +the other side.) \end{funcdesc} \begin{funcdesc}{cos}{x} @@ -45,10 +75,13 @@ Return the exponential value \code{e**\var{x}}. \begin{funcdesc}{log}{x} Return the natural logarithm of \var{x}. +There is one branch cut, from 0 along the negative real axis to +-\infinity, continuous from above. \end{funcdesc} \begin{funcdesc}{log10}{x} Return the base-10 logarithm of \var{x}. +This has the same branch cut as \function{log()}. \end{funcdesc} \begin{funcdesc}{sin}{x} @@ -61,6 +94,7 @@ Return the hyperbolic sine of \var{x}. \begin{funcdesc}{sqrt}{x} Return the square root of \var{x}. +This has the same branch cut as \function{log()}. \end{funcdesc} \begin{funcdesc}{tan}{x} @@ -89,3 +123,18 @@ and perhaps don't even know what they are. They would rather have Also note that the functions defined in \module{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: + +\begin{seealso} + \seetext{Kahan, W: Branch cuts for complex elementary functions; + or, Much ado about nothings's sign bit. In Iserles, A., + and Powell, M. (eds.), \citetitle{The state of the art in + numerical analysis}. Clarendon Press (1987) pp165-211.} +\end{seealso} |