A mean value property of harmonic functions on the interior of a hyperbola
Let \(Y=\bigl \{[x,y]\in \mathbb R^2\:x>0,x^2/a^2 -y^2 /b^2>1\bigr \}\), \(c=\sqrt {a^2 +b^2}\). For a bounded function \(h\), which is harmonic on a neighbourhood of the closure of \(Y\), the identity NEWLINE\[NEWLINE\int _c^\infty h(x,0)/\sqrt {x^2 -c^2}\;{\operatorname {d}}x=\int _{-\infty }^\infty h(a \cosh r, b \sinh r)/2\;{\operatorname {d}}rNEWLINE\]NEWLINE is established. The method of the proof of this identity is solving the Dirichlet problem for \(Y\) by separation of variables.
scientific article; zbMATH DE number 6086824