Contour integration is a method of evaluating integrals of functions along oriented curves in the complex plane. It is an extension of the usual integral of a function along an interval in the real number line. Contour integrals may be evaluated using direct calculations, the Cauchy integral formula, or the residue theorem.
A curve in the complex plane is a set of points parameterized by some function , usually denoted parameterized by . is differentiable if exists for all .
Let and be a differentiable curve parameterized by . The integral of along is defined as
If is a holomorphic function, then one may think of it as a function from a subset of to the whole . In other words, implicitly defines a vector field on . Accordingly, one naturally integrates holomorphic functions in the same way as one integrates vector fields, using line integrals.
Some contour integrals may be computed by hand. For example:
Let denote the circle centered at with radius . Then compute
First, parameterize the circle by , so that as ranges over , the entire circle is traced out once. Note that ; a curve with this property is called closed.
With this parameterization, one computes
Note that in the above example, the function is not holomorphic on the region interior to , since it is not defined at .
Take to be holomorphic. The primitive of is a function such that for all .
Suppose is a curve parametrised by and has a primitive. Then, since complex differentiation follows the chain rule,
In particular, if is a closed curve, i.e. , then . There is a result that all holomorphic functions have primitives. Thus, the only nontrivial contour integrals along closed curves are those which enclose poles of the function being integrated.