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Date: Around 1995

How can you show that an integral is non-elementary?

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?


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 closed-form 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.


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