**Question.** Let $f: \mathbf{R} \to \mathbf{R}$ be an analytic function. Is there a harmonic function $u$ on the circular cylinder $D \times \mathbf{R} \subset \mathbf{R}^3$ so that $u = f$ along the axis $\{ (0,0) \} \times \mathbf{R}$?


* The problem is obviously ill-posed in the sense of Hadamard because it is very underdetermined. Although it doesn't make sense as a 'Dirichlet problem', I think a (positive or negative) answer is possible.
* Probabilistic arguments seem tricky because the axis is too small for hitting times of Brownian motion to be defined.



'Dirichlet problem' along axis for harmonic functions
Assuming the Taylor series of $f$ has an infinite radius of convergence, the sum
$$
\sum\_{k=0}^\infty \left(x^2+y^2\right)^kf^{(2k)}(z)\cdot \frac{(-1)^k}{4^k k!^2}
$$
converges absolutely and locally uniformly on $\mathbb{R}^3$ to a function $u(x,y,z)$ which is harmonic and satisfies $u(0,0,z)=f(z)$


