Analytic
Functions| Analytic
Functions |
Holomorphic Function > s.a. Meromorphic.
* 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 ab 
a f =
i b f ,
where J is a complex structure.
* Applications: Complex
transformations in electromagnetism; Segal-Bargmann
transform in quantum mechanics (> see coherent states).
@ References: Hall qp/99-ln
[in theoretical physics]; Zhu 04 [in the unit ball].
Analytic Functions and Mappings > s.a. conformal
transformations.
$ Cauchy theorem: Given a complex function f, for all contours C which
are homotopically trivial in the domain of analyticity of f,
C f(z)
dz = 0 ;
More generally, if the only singularities inside C are isolated poles
of f, the integral is equal to 2
times
the sum of the residues at those points.
* Schwarz transformation:
A map f : C → 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].
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
21 jun 2008