Analytic Functions| 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 ab ∇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.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 2 jul 2018