Let x be a random variable distributed according to the density
function 
such that 
. We
define the moment generating function M(T):
and the cumulant generating function 
. To restore
we can apply the Fourier inversion formula as follows:
Let 
be the mean of n independent x's. Its density
function 
is therefore:
By Cauchy's theorem the value of the integral is unchanged if we integrate along any line parallel to the imaginary axis, i.e.
To evaluate the integral we will expand the contents of the
exponential as a power series in y about 
.
Then:
Next, we separate the first two terms in the exponential, and then expand the remaining terms as a product of power series:
Multiplying out the product of summations, we get:
We now apply the standard integral, true for 
:
where 
is the Hermite polynomial of degree j. Thus:
Note 
, 
. It is not
clear that the first term dominates either the highest or lowest order
contributions to 
(the formal proof of convergence requires a
number of other results). However, when 
, the
arguments of the Hermite functions become zero, thus all odd orders
disappear and only the constant terms of even orders contribute. It
is certainly reasonable that this should be the best estimate to the
integral.
The approximation
when 
is chosen such that 
is called the
Saddlepoint approximation to 
.