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From: madler@piglet.caltech.edu (Mark Adler)
Newsgroups: comp.sys.handhelds
Subject: Re: Question - Indefinite Integration
Message-ID: <1990Sep28.182551.14436@nntp-server.caltech.edu>
Date: 28 Sep 90 18:25:51 GMT
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I responded to the e^(y^2/2) integration question privately, but it seems
others are curious ...

First off, what he really wanted was e^(-y^2/2), which is rather different
for real y, and also quite useful, being the gaussian distribution (up to
normalization).  There is no symbolic answer that can be expressed as a
rational function of trig and/or log/exp functions (except for special
values of the integration bounds like -infinity, 0, and infinity).

What is done when you have a useful, unsolvable integral is "solve it by
naming it".  The integral is given a name (the error function or
complementary error function, erf(x) or erfc()), and then you have tables
and subroutines to evaluate it.  In fact, the HP has a function that'll
do it: UTPN (upper tail probability normal) in MATH/PROB.  Given the
arguments 0 1 x, UTPN will return the integral from x to infinity of
e^(-t^2/2)/sqrt(2pi) dt.  The sqrt(2pi) is to make the result one for
x equal to -infinity.  The complementary error function (erfc()) is
a little different, but a simple transformation of UTPN will give it.

Naturally, UTPN is much faster (and probably more accurate) than
doing the integral numerically on the HP-48.

Mark Adler
madler@piglet.caltech.edu