Watson's lemma [SEP] named after
Watson's lemma
In mathematics, Watson's lemma, proved by G. N. Watson (1918, p. 133), has significant application within the theory on the asymptotic behavior of integrals.
Statement of the lemma.
Let formula_1 be fixed. Assume formula_2, where formula_3 has an infinite number of derivatives in the neighborhood of formula_4, with formula_5, and formula_6. 
Suppose, in addition, either that 
where formula_8 are independent of formula_9, or that
Then, it is true that for all positive formula_11
each integer formula_43 we have thus shown that
as formula_42, where the infinite series is interpreted as an asymptotic expansion of the integral in question.
Example.
When formula_75, the confluent hypergeometric function of the first kind has the integral representation
where formula_55 is the gamma function. The change of variables formula_78 puts this into the form
which is now amenable to the use of Watson's lemma. Taking formula_80 and formula_81, Watson's lemma tells us that
which allows us to conclude that
References.