Complexification Of The Integrand

Complexification of the Integrand:

We evaluate

by using the contour . The obvious choice for the complexification of the integrand is

Now becomes quite large on when R is large and positive. There is no evident way to alter the contour so that good estimates result. Instead, we alter the function! Let

On the one hand (for R > 1),

On the other hand, with

And

Hence we get;