n the calculation of the section
(
Asymptotic
expansion of Laplace integral
) we want to place the limits of integration
so that the difference term of the integration by parts would grab the most
significant value. To be precise, let us consider the
integral
and assume that the function
has exactly one local maximum on the interval
at point
.
We assume that
is continuously differentiable and
is twice continuously differentiable,
thus,
We cannot directly apply the same technique of the section
(
Asymptotic
expansion of Laplace integral
)
because
and we have a pole in the denominator. Instead, we do the following change of
variable.
There is a neighborhood
where
and
for
.
In such neighborhood we introduce a variable
according to the
relationship
In addition, we introduce the inverse relationship functions
:
Then
Next, we investigate the behavior of
around
(and drop the notation
):
where the numerator and denominator vanish as
.
Thus, at
where
is negative.
Then
Whether the
is positive or negative depends on orientation of variable
.
Consider
:
if
increases from
to
then
becomes negative and we may have
or
.
We choose
.
We
have
and we apply the proposition
(
Asymptotic of
integral with Gaussian
kernel
),
We summarize our findings.
