For , let be a complex domain and let . Let be a complex Brownian motion starting from . Set .

Suppose that there exists a conformal isomorphism such that . Set and define for

and .

Then and have the same distribution.

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complex analysis

# Too complex to be true but too symmetrical to be false

# It’s a matter of principles (of convergence)

# Convergence is contagious

For , let be a complex domain and let . Let be a complex Brownian motion starting from . Set .

Suppose that there exists a conformal isomorphism such that . Set and define for

and .

Then and have the same distribution.

Let be a sequence of entire functions such that, for every , there exists . Suppose that is continuous and . Let be an entire function such that . Prove that over compact sets if and only if , for every .

Suppose is function and that at a point such function is analytic. If the infinite sum converges, then in fact is entire and such sum converges for any other choice of .