Riemann mapping theorem
The
Riemann mapping theorem in
complex analysis states the following: if
U is a
simply connected open subset of the
complex number plane C (or even of the compact complex number sphere
C U {∞}) which is different from
C (and
C U {∞}),, then there exists a
bijective holomorphic conformal map f :
U -> D, where
D = {
z in
C : |
z| < 1 } denotes the open disk. Intuitively, the condition that
U be simply connected means that
U does not contain any "holes"; the conformality of
f means that
f maintains the shape of small figures.
The map f is essentially unique: if z0 is an element of U and φ in (-π, π] is an arbitrary angle, then there exists precisely one f as above with the additional properties f(z0) = 0 and arg f '(z0) = φ.
As a corollary, any two such simply connected open sets (which are different from C and C U {∞}) can be conformally mapped into each other.
The theorem was proved by Bernhard Riemann in 1851, but his proof depended on a statement in the calculus of variations which was only later proven by David Hilbert.