 Analytic Functions

Holomorphic Function > s.a. Meromorphic Function.
* Idea: A complex function f(z) = u(x,y) + i v(x,y), where z = x + i y and u and v are real functions satisfying the Cauchy-Riemann conditions,

x u = ∂y v   and   ∂y u = −∂x v ,   or   J aba f = i ∇b f ,

where J is a complex structure.
* Applications: Complex transformations in electromagnetism; Segal-Bargmann transform in quantum mechanics; The Segal-Bargmann transform or heat transform maps a square-integrable function on a Euclidean space to a solution to the heat equation, and this solution can be extended holomorphically to the complexification (> s.a coherent states).
@ References: Hall CM(00)qp/99-ln [in theoretical physics]; Zhu 04 [in the unit ball]; Olafsson 13 [Segal-Bargmann transform and Hilbert spaces of holomorphic functions].

Analytic Functions and Mappings > s.a. Argument Principle; conformal transformations.
\$ Cauchy theorem: Given a complex function f, for all contours C which are homotopically trivial in the domain of analyticity of f,

$$\textstyle \oint_C f(z)\, {\rm d}z = 0\; ;$$

More generally, if the only singularities inside C are isolated poles of f, the integral is equal to 2π times the sum of the residues at those points.
* Schwarz transformation: A map f : $$\mathbb C$$ → $$\mathbb C$$ which is analytic except at a finite set of points, and maps a polygon to the real line.
@ General references: Ahlfors 53; Cirelli & Gallone 73; Evgrafov 78.
@ Applications: in Panofsky & Phillips 62 [Schwarz, in electromagnetism]; Krantz AS(99)#5 [conformal mappings].