The methods of integration of complex functions and their underlying theories are discussed in this chapter. The cornerstones in complex integration are the Cauchy–Goursat theorem and the Cauchy integral formula. A fascinating result deduced from the Cauchy integral formula is that if a complex function is analytic at a point, then its derivatives of all orders exist and these derivatives are analytic at that point. Other important theorems include Gauss' mean value theorem, Liouville's theorem, and the maximum modulus theorem.

Many properties of the complex integrals are very similar to those of the real line integrals. For example, when the integrand satisfies certain conditions, the integral can be computed by finding the primitive function of the integrand and evaluating the primitive function at the two end points of the integration path. However, there are other properties that are unique to integration in the complex plane.

In the last section, we link the study of conservative fields in physics with the mathematical theory of analytic functions and complex integration. The prototype conservative fields considered include the gravitational potential fields, electrostatic fields and potential fluid flow fields. The potential functions in these physical models are governed by the Laplace equation, and so their solutions are harmonic functions. Complex variables techniques are seen to be effective analytical tools for solving these physical models.