solving 2D linear Klein-Gordon equation for infinite domain Anyone knows how to solve 
$$ v_{tt}=c^2\Delta v - m^2 v, x \in R^2 $$
with 
$v(x,0)=g(x)$ and $v_t(x,0)=h(x)$.
Many thanks
 A: Using the Fourier transform you will get an ODE of second order, likely
\begin{equation}
y''=\alpha c^2x^2y-m^2y,
\end{equation}
where $\alpha$ is some constant related to Fourier transform. This is a linear ode, then you can transfer this to an ode system of first order, and you can calculate the solution with undetermined coefficients. The solution is the Fourier transform, the inverse formula gives you the real solution. Now the coeffients can be determined by the boundary conditions.
A: Hint:
Let $p=\dfrac{mx}{c}$ ,
Then $v_{tt}=m^2v_{pp}-m^2u$ with $v(p,0)=g\left(\dfrac{cp}{m}\right)$ and $v_t(p,0)=h\left(\dfrac{cp}{m}\right)$
Let $q=mt$ ,
Then $u_{qq}=u_{pp}-u$
Similar to Deducing the existence of a PDE by constructing it inductively via its Taylor series expansion,
Consider $v(p,0)=G(p)$ and $v_q(p,0)=H(p)$ ,
Let $v(p,q)=\sum\limits_{m=0}^\infty\dfrac{q^m}{m!}\dfrac{\partial^mu(p,0)}{\partial q^m}$ ,
Then $v(p,q)=\sum\limits_{m=0}^\infty\dfrac{q^{2m}}{(2m)!}\dfrac{\partial^{2m}u(p,0)}{\partial q^{2m}}+\sum\limits_{m=0}^\infty\dfrac{q^{2m+1}}{(2m+1)!}\dfrac{\partial^{2m+1}u(p,0)}{\partial q^{2m+1}}$
$v_{qqqq}=v_{ppqq}-v_{qq}=v_{pppp}-v_{pp}-v_{pp}+v=v_{pppp}-2v_{pp}+v$
$v_{qqqqqq}=v_{ppppqq}-2v_{ppqq}+v_{qq}=v_{pppppp}-v_{pppp}-2v_{pppp}+2v_{pp}+v_{pp}-v=v_{pppppp}-3v_{pppp}+3v_{pp}-v$
Similarly, $\dfrac{\partial^{2n}v}{\partial q^{2n}}=\sum\limits_{k=0}^n(-1)^{n-k}C_k^n\dfrac{\partial^{2k}v}{\partial p^{2k}}$
$v_{qqq}=v_{ppq}-v_q$
$v_{qqqqq}=v_{ppqqq}-v_{qqq}=v_{ppppq}-v_{ppq}-v_{ppq}+v_q=v_{ppppq}-2v_{ppq}+v_q$
$v_{qqqqqqq}=v_{ppppqqq}-2v_{ppqqq}+v_{qqq}=v_{ppppppq}-v_{ppppq}-2v_{ppppq}+2v_{ppq}+v_{ppq}-v_q=v_{ppppppq}-3v_{ppppq}+3v_{ppq}-v_q$
Similarly, $\dfrac{\partial^{2n+!}v}{\partial q^{2n+1}}=\sum\limits_{k=0}^n(-1)^{n-k}C_k^n\dfrac{\partial^{2k+1}v}{\partial p^{2k}\partial q}$
$\therefore v(p,q)=\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n\dfrac{(-1)^{n-k}C_k^nG^{(2k)}(p)q^{2n}}{(2n)!}+\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n\dfrac{(-1)^{n-k}C_k^nH^{(2k)}(p)q^{2n+1}}{(2n+1)!}$
