In Richard Feynman's book, *Surely You're Joking Mr. Feynman!*, he says:

One time I boasted, "I can do by other methods any integral anybody else needs contour integration to do."

So Paul [Olum] puts up this tremendous damn integral he had obtained by starting out with a complex function that he knew the answer to, taking out the real part of it and leaving only the complex part. He had unwrapped it so it was only possible by contour integration! He was always deflating me like that. He was a very smart fellow.

There is a similar question on the site about this, but I'd like to know how one can create an integral that can only be evaluated using contour integration. Can someone elaborate on what Paul Olum did to "unwrap" the integral and leave only the complex part?