Every solution of $\Delta\phi+\phi=0$ on $\mathbf R^2$ can be written as the “Poisson” integral ($(u,v)=w$, $(x,y)=z$)
$$
\phi(z)= \left\langle T, e^{i\langle z,\cdot\rangle}\right\rangle=\int_{\mathrm S^1}e^{i\langle z,w\rangle}dT(w)
$$
for a unique ***entire functional*** $\,T$ on the circle; these include Radon measures, Schwartz distributions, hyperfunctions, and more. Details and proofs see Hashizume et al. ([1972](//ams.org/mathscinet-getitem?mr=55:3168), p. 543), Helgason ([1974](//ams.org/mathscinet-getitem?mr=51:3353), p. 348; [1984](//ams.org/mathscinet-getitem?mr=86c:22017), p. 5), or Agmon ([1999](//ams.org/mathscinet-getitem?mr=2000k:35043)). For example, if I am not mistaken,

- $T_1=$ Dirac measure at $(-i,\sqrt2)\in (\mathrm S^1)^{\mathbf C}$ gives $\phi_1(z)=e^{x+\sqrt2iy}$ (found by Maple);
- $T_2=$ derivative of $T_1$ in the direction $(\sqrt2,i)$ gives $\phi_2(z)=(\sqrt2x+iy)e^{x+\sqrt2iy}$ (new).