ABSTRACT
The complex analytic functions on an open set form a vector space, because sums and constant multiples of analytic functions are analytic. This chapter discusses the basic material on complex line integrals and utilizes Cauchy theory. It describes complex line integrals to evaluate real integrals that arise in applications. The Cauchy-Riemann equations lead, via Green's theorem, to a proof of the Cauchy theorem. In complex analysis courses, one can prove this result directly without passing through Green's theorem. A convergent power series defines a complex analytic function inside the region of convergence. One of the reasons why complex variable theory is so useful in science is that complex variable methods enable us to evaluate many integrals rather easily. Complex analysis, especially the residue theorem, enables us to evaluate many line integrals.
