Asymptotic expansion of integral with Gaussian kernel.

roposition

(Asymptotic of
integral with Gaussian kernel) Suppose for
the function
may be
expanded
and for some
the
integral
exists. Then for
the following expansions
hold

Proof

Split the integral
into
two
pieces
for some number
For the second integral we
estimate

We substitute the expansion for
into the first integral
We make a change
,
,
.
At this point we bring in the gamma
function

(Gamma function primer)

For the first term we
have
We evaluate the second
term
We evaluate the third
term
We collect the
results
It remains to note that the term
disappears from expansion of
by symmetry
.