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Methods of Mathematical Physics, Third Edition by Bertha Jeffreys, Harold Jeffreys

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Chapter 17ASYMPTOTIC EXPANSIONS

Up the airy mountain  Down the rushy glen.

WILLIAM ALLINGHAM

17·01. Nature of asymptotic expansions: incomplete factorial function.

The incomplete factorial function is defined by the integral

image

where x and n are positive. Integrate by parts; we have

image

This is exact. Now the integral in the last term is always positive, but its coefficient alternates in sign with successive values of r. Consequently the error committed in neglecting it alternates in sign, and therefore I always lies between the sums of r and r+1 terms ...

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