## Sleek proof that the harmonic series diverges

Suppose that the harmonic series does converge. Then the sequence of integrable functions functions is bounded above by an integrable function .

## Problem regarding subharmonic functions

Suppose that a function of class satisfies the inequality at every point of . Suppose also that all its critical points are non-degenerate, i.e. the matrix of second order derivatives at the critical point has non-zero determinant. Prove that cannot have local maxima.

## Weakly harmonic implies harmonic

Suppose is weakly harmonic on an open set , i.e. the relation holds for all . Show that is harmonic in .

PHD Iowa (6202)

See this blog post for a proof.

## Divergent integral

Let satisfy on . Show that the integral is convergent if and only if .

*PHD 6201*

## Harmonic functions and holomorphic functions

Prove that a function is harmonic if and only if there exists an entire function such that . ( if and only if it is the real part of an entire function)

## Harmonic function

Let be semilines on the plane starting from a common point. Prove that if there doesn’t exist any function harmonic on the whole plane that vanishes on the set , then there exists a pair such that there is no function harmonic on the whole plane such that vanishes on .

*Miklos Schweitzer 2001*

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