Cauchy's Integral Theorem
Cauchy's integral theorem allows for integration over a complex variable. It deals only with analytic functions, that is, those functions which have only a single value at each point in the region of interest. These functions have the same limits on a point no matter which direction that point is approached. The analyticity of a function can be tested with the Cauchy-Riemann equations, which relate the first derivatives of the real and imaginary components of a function in terms of the real and imaginary components of the variable. If a function is analytic, Cauchy's integral theorem states that the integral between two points is always the same regardless of what path is taken.
To state Cauchy's theorem another way, any closed curve in an analytic region integrates to zero. This means that a function can be integrated around a circle, a square, or a totally arbitrary closed curve and always get the same result: zero.
Cauchy's integral theorem also deals with functions which are not analytic, usually because of a discontinuity caused by attempting to divide by zero somewhere in the function's range. In this case, the result of the integration around the closed curve is 2 times pi times the square root of negative one (also denoted as i), times the residue of the function at the nonanalytic point. (The point of non-analyticity is sometimes called a pole.) The residue is calculated by decomposing the function into an analytic and a non-analytic part, and then taking the value of the analytic part at the pole.
This theorem may also cover situations where an entire half of the Cartesian plane--positive or negative values of the real or imaginary component--may be the integral surface. In that situation, all poles and discontinuities in the half-plane are accounted for, and the curve used must disappear at the appropriate (positive or negative) infinity.
This theorem is often used in physics and engineering applications. Most of the real-world functions scientists and engineers deal with work in at least two dimensions. Quantum mechanics especially requires many integrals in a two-dimensional coordinate system whose axes can be thought of as corresponding with real and imaginary numbers. Also, many quantum mechanical problems require an integration out to infinity to take into account all of space and time. In this case, Cauchy's integral theorem is one of the best ways to solve the problem, even if the original idea is formulated so that it's only one-dimensional. Taking an integral out to infinity and accounting for divisions by zero are the theorem's most useful features.
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