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Abstract:
A complex function f: C → C is
said to be holomorphic at a point z0 if its derivative f'(z)
exists in a neighbourhood of z0. A complex function holomorphic in
the whole complex plane is called an entire function. Picard's Little
theorem says something remarkable about entire functions: if f'(z) is
an entire function and f(C) misses two points (or more),
then f is in fact a constant function. We will give a non-standard
proof of this fact using basic differential geometry, which will also
hopefully reveal why a non-constant entire function can miss one
complex number but not two. This proof was casually noted as a simple
observation by the great analyst Lars Valerian Ahlfors around 1936. |