
> 
> Is there an explicit formula for the Fourier transform of the generalized function of 2 variables
>  $$\frac{1}{x+y^2+i0}?$$
> 
> 
> 


Remark. **Equivalent question:** consider the Schroedinger equation one the line
$$i\frac{\partial}{\partial t}\Psi(x,t)=\frac{\partial^2}{\partial x^2}\Psi(x,t).$$



> 
> Find the kernel $K(x,t)$ such that for $t>0$
>  $$\Psi(x,t)=\int\_{-\infty}^\infty K(x-y,t)\Psi(x,0)dy.$$
> 
> 
> 



Fourier transform of a generalized function on the plane
The Fourier transform vanishes for $u>0$, for $u<0$ instead
$$
I(u,v)=\frac{1}{2\pi}\int\_{-\infty}^\infty dx\int\_{-\infty}^\infty dy\,\frac{e^{\mathrm{i}(ux+vy)}}{x+y^2+\mathrm{i}0^+}$$
$$\qquad\qquad=-ie^{i\pi/4}\sqrt{\pi }(- u)^{-1/2}\exp\left({\frac{i v^2}{4 u}}\right)\;\;\text{for}\;\; u<0. 
$$
For $u=0$ there is a delta function, $I(0,v)=-i\pi\delta(|v|)$. Note that  

$$\int\_{-\infty}^\infty I(u,v)\,dv=\begin{cases}
0&\text{if}\;\;u>0,\\
-\pi i&\text{if}\;\;u=0,\\
-2\pi i&\text{if}\;\;u<0.
\end{cases}$$


