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Shift the contour of integration

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Quick description

When integrating a holomorphic (or meromorphic) function f(z) on some contour \gamma in the complex plane (or even just on an interval in the real line, such as [-R,R]), bear in mind that Cauchy's integral theorem and its variants (such as the Cauchy integral formula and the residue theorem are available to shift the contour \gamma of integration to a different contour \gamma' with the same endpoints; the above theorems will, in many cases, guarantee that the two contour integrals are the same, except possibly for some correction terms coming from poles or other singularities between \gamma and \gamma'.

This technique is often good for bounding integrals, because one can sometimes shift the contour into regions where the integrand (and hence the integral) is quite small. In some cases, one can even shift the contour all the way to infinity by an appropriate limiting process, which can allow one to use complex methods to evaluate a number of improper integrals exactly.

Prerequisites

Complex analysis

Example 1

(Maybe do \int_{-\infty}^\infty \frac{e^{ix}}{x^2+1}\ dx in detail? Note that Jordan's lemma is already discussed at base times height and Bound by a Riemann sum.)

General discussion

It is important to note that contour shifting methods generally only work when the integrand f can be expressed as a holomorphic (or at least meromorphic) function of the variable z of integration. In particular, integrands involving expressions such as |z| or \overline{z} tend to not be amenable to contour integration techniques, unless the integrand can somehow be rewritten to eliminate such expressions.

A classical application of the contour shifting method to bound integrals is the method of steepest descent.

See also "If you are calculating an integral, complex substitutions are not forbidden".