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*^{ a}_{b}
∇_{a} *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*,

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].

> __Online resources__:
see MathWorld page;
Wikipedia page.

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send feedback and suggestions to bombelli at olemiss.edu – modified 2 jul 2018