MathDB

Let the complex function

F(z)

be regular on the punctuated disk

\{ 0<|z| < R\}

. By a level curve we mean a component of the level set of

\mathrm{Re}F(z)

, that is, a maximal connected set on which

\mathrm{Re}F(z)

is constant. Denote by

A(r)

the union of those level curves that are entirely contained in the punctuated disk

\{ 0<|z|<r\}

. Prove that if the number of components of

A(r)

has an upper bound independent of

r

then

F(z)

can only have a pole type singularity at

0

Problem Statement