integrals with branch points

Up to now the contour methods assumed the integrand was single-valued — rational functions, exponentials. But many important real integrals contain a fractional power x^s, a square root, or a logarithm log x, and these are inherently multivalued: as you walk around the origin once, x^s comes back multiplied by e^(2 pi i s) and log x comes back increased by 2 pi i. You cannot integrate such a function around a closed loop unless you control that multivaluedness. This is the whole genre of branch-cut integrals.

The unifying idea is to draw a branch cut — a curve the contour is forbidden to cross — that pins the multivalued function down to a single, definite branch. Then you choose a contour that hugs the cut and exploits the function's jump across it. Two standard shapes do most of the work: the keyhole contour, for a single branch point with the cut running off to infinity (typical for x^(s-1) f(x) on [0, infinity) and for log-weighted integrals), and the dogbone contour, for two branch points joined by a finite cut (typical for a square root of a quadratic on a finite interval). In both, the top and bottom edges of the cut carry different values of the function, so their contributions combine into a known multiple of the target integral, and the residue theorem closes the deal.

A recurring bonus trick: inserting an extra factor of log z into an otherwise rational integrand and integrating around a keyhole often computes an integral that had no logarithm at all, because the jump in log z across the cut isolates the wanted real integral. The honest caution is that everything here depends on fixing the branch consistently around the entire contour; a single sloppy choice of arg z makes the jump wrong and the answer meaningless. Branch-cut integration is powerful precisely because it turns multivaluedness from an obstacle into the engine.