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Subject: \hspace*{0.25in}\=More mathematical functions\\
From: \>Van Snyder\\
Reference: \>03-258r1, section 2.4.4.3\\
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\section*{Number}
TBD
\section*{Title}
More mathematical functions.
\section*{Submitted By}
J3
\section*{Status}
For consideration.
\section*{Basic Functionality}
More mathematical functions.
\section*{Rationale}
Mathematical functions for complex type are occasionally needed. The
only ones that are available for complex type are ABS, COS, EXP, LOG and
SIN. The other mathematical functions that are provided for real type are
useful in practice for complex type as well. Inverse hyperbolic
functions and other functions are useful.
\section*{Estimated Impact}
Minor but tedious.
\section*{Detailed Specification}
Provide ACOS, ASIN, ATAN, COSH, SINH, TAN and TANH for complex type.
Provide inverse hyperbolic functions, including for complex type.
The following also appear in applications, and have better round-off
characteristics for $x$ near zero when implemented directly rather than
as written here: $e^x-1$, $\log(x+1)$, $x-\log(x+1)$,
$(x-\sin(x))/x^3$, $(1-\cos(x)/x^2$, $(\sinh(x)-x)/x^3$,
$(\cosh(x)-1)/x^2$ and $1/\Gamma(x+1)-1$. The function $x-1-\log(x)$ has
better round-off characteristics for $x$ near one when implemented
directly rather than as written here. These should be provided for both
real and complex arguments. The first two are the ones most commonly
found in applications.
A few other functions are useful, especially $\Gamma(x)$, erf$(x)$,
erfc$(x)$ and $\exp(x^2)$ erfc$(x)$.
These are sufficiently difficult to do well for complex arguments that
the standard should not require it.
\section*{History}
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