

Nonelementary integrals
Name: existing
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1995
Question:
How can you show that an integral is nonelementary?
Replies:
I suppose you mean, how can you show that an integral cannot be evaluated in
terms of a finite sum of known functions? Is that what you mean? Suppose
you had such an integral. Can you not then define a new function that is
equal to that integral, explore the properties of this new function, and
write a paper for a mathematics journal describing the function's properties
and suggesting that the function be named after you? Did I answer your
question?
jlu
Well, there is a standard approach used by people who really want to know
whether they have to evaluate an integral numerically or whether there is
some closedform solution in terms of functions whose properties we are very
familiar with (usually for computational purposes this means having a
formula that gets 16 or so digits in under 10 floating point operations).
The approach is to try all the standard substitutions and other simplifying
techniques to get the integral into all its "simplest" forms (usually an
integral has 2 or three different representations with comparable complexity
 i.e. number of different factors, distinguished according to type, such as
rational functions, algebraic functions, exponential functions, trigonometry
functions, etc.). Giving the integral in these various standard forms (and
with whatever parameters it may have removed as much as possible by being
taken outside the integral somehow) you then try to find it in one of the
big tables of integrals, such as that compiled by the Russians Gradshteyn
and Ryzhik. If you and other competent colleagues cannot do this after a
couple of days work, it is probably not possible.
asmith
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Update: June 2012

