The first four chapters cover the essential core of complex analysis presenting their fundamental results. However, a much richer set of conclusions can be drawn about a complex analytic function than is generally true about real di erentiable functions. Lecture notes for complex analysis lsu mathematics. Find all complex solutions of the following equations. A function f z is said to be analytic at a point z if z is an interior point of some region where fz is analytic. Complex numbers, functions, complex integrals and series. We also develop the cauchyriemannequations, which provide an easier test to verify the analyticity of a function. Preliminaries to complex analysis 1 1 complex numbers and the complex plane 1 1. A brief introduction to complex functions, including basics and holomorphicity, as well as comparisons to real functions. A function f z is said to be analytic in a region r of the complex plane if fz has a derivative at each point of r and if fz is single valued. This is an extremely useful and beautiful part of mathematics and forms the basis of many techniques employed in many branches of mathematics and physics. We also show that the real and imaginary parts of an analytic function are solutions of the laplace equation. After this standard material, the authors step forward to elliptic functions and to elliptic modular functions including a taste of all most beautiful results of this field.