Free Variations on the (eternal) Theme of Analytic Continuation

“Between two truths of the real domain, the easiest and shortest path quite often passes through the complex domain.”                               P. Painlevé, 1900



– When does the Taylor series $\sum a_nz^n $ represent a rational, or an algebraic function? Why does the Taylor series $\sum\cos\sqrt{n}z^n$ extend to the whole complex plane except for the point 1, while $ \sum 2^{-n}z^{n^2}$ does not extend anywhere beyond the unit circle?
– How far does the Newtonian potential of a solid (or, the logarithmic potential of a plate) bounded by an algebraic surface (curve)extend inside the solid? How come the singularities of such potential are algebraic for an ellipse and an oblate spheroid and transcendental for a prolate spheroid?
– How does one find singularities of an axially symmetric harmonic function in the ball from the coefficients in its expansion in spherical harmonics?
– If a line intersects a spherical shell over two disjoint segments and a harmonic function in the shell vanishes on one, does it have to vanish on the other one?
– Where does the solution of the Dirichlet problem in a domain with algebraic boundary might have a singularity outside the domain?
We shall discuss these questions in the unified light of analytic continuation, and, in particular, analytic continuation of solutions to analytic PDE.