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The Dirichlet problem. Let be open in C and f : C. f is holomorphic in if any (hence, all)
 

Summary: The Dirichlet problem.
N.A.
13/2/07
Let be open in C and f : C. f is holomorphic in if any (hence, all)
of the following properties hold:
(i) f has a complex derivative at any point z ,
f (z) = lim
h0
f(z + h) - f(z)
h
.
(ii) u = Re(f) and v = Im(f) satisfy Cauchy-Riemann's equations:
ux = vy
uy = -vx
If we think of f : R2
R2
as a function between Euclidean planes, the
CR equations say that the Jacobian of f has the particular form
Jf(x, y) =
a -b

  

Source: Arcozzi, Nicola - Dipartimento di Matematica, UniversitÓ di Bologna

 

Collections: Mathematics