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, p. 543), Helgason (1974, p. 348; 1984, p. 5), or Agmon (1999). 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).

Every solution of $\Delta\phi+\phi=0$ on $\mathbf R^2$ can be written as the “Poisson integral” (where for short $(u,v)=w$, $(x,y)=z$) $$ \phi(z)= \left\langle T, e^{i\langle z,\cdot\rangle}\right\rangle=\int_{\mathrm S^1}e^{i\langle z,w\rangle}dT(w) $$ for a unique entire functional $\,T$ on the circle $\mathrm S^1$; these include Radon measures, Schwartz distributions, hyperfunctions, and more. Details and proofs see Hashizume et al. (1972, p. 543), Helgason (1974, p. 348; 1984, p. 5), or Agmon (1999). 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).