Study

Theory of functions of a complex variable is the study of holomorphic functions. Holomorphic functions are the functions most like polynomials. They are the “polynomials of finite or infinite degree” with complex number coefficients. “Infinite degree” means “power series”, with the restriction that we only use power series at places where they are convergent.

Complex analysis is one of the most beautiful of all subjects because its functions have so many nice properties and because it interacts in deep ways with so many other branches of mathematics and physics.  Contour integration, for example, provides a method of computing difficult integrals. Another example is Picard’s great theorem, which states that an analytic function assumes every complex number, with possibly one exception, infinitely often in any neighborhood of an essential singularity.