Check out Griffiths' Introduction to Quantum Mechanics, around page 320 or 330 (I don't have my copy with me). There's an interesting contour integral there that takes up about two pages of analysis.
Does anyone have any examples of interesting real-valued definite integrals that can be evaluated by contour integration that they like to share? Maybe one with an unusual contour, or perhaps one with an essential singularity (i.e, something not prototypical).