I Solving the Wave Equation via complex coordinates

[jk22]

I'm looking for material about the following approach : If one suppose a function over complex numbers $f(x+iy)$ then

$\frac{df}{dz}=\frac{\partial f}{\partial x}\frac{1}{\frac{\partial z}{\partial x}}+\frac{\partial f}{\partial y}\frac{1}{\frac{\partial z}{\partial y}}=\frac{\partial f}{\partial x}-i \frac{\partial f}{\partial y}$

Hence the wave equation with source g reads

$Re(\frac{\partial^2 f}{\partial z^2})=g(x+iy)$

Using Cauchy residue theorem

$Im(\oint\frac{f(z)}{(z-a)^3}dz)=\pi g(a)$

Then if the source is the function itself we can obtain the total function out of the contour integral over the boundary condition for example.

However I seek to use this method to solve the wave equation given another source and don't know how to solve the integral equation. Does anyone know how this could be done ?

 

[hunt_mat]

You can write the Laplacian in terms of complex co-ordinates, I'm not sure you can do this for the wave equation though, as you have a minus sign to contend with.

 

[jk22]

Ok, I have to check, but is it correct that gives the solution of the Klein-Gordon equation with a quantization due to the winding number ?

 

[jk22]

Oops, I made mistakes.

1) $\frac{d}{dz}=\frac{1}{2}(\frac{\partial}{\partial x}-i\frac{\partial}{\partial y})$

I forgot also that $y=ct$

Then we have to derive twice

$f(z)=\frac{n_\gamma(z)}{2\pi i}\oint_\gamma\frac{f(s)}{z-s}ds$

Towards $z$ :

$\frac{d^2}{dz^2}=\frac{\partial^2}{\partial x^2}-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}-2i\frac{\partial}{\partial x}\frac{\partial}{c\partial t}$

This seems more clear to ask the questions :

1) is the product of operator derivative a second order derivative or the square of the first derivative : $\frac{\partial^2\psi(x,t)}{\partial x^2}\neq\left(\frac{\partial\psi(x,t)}{\partial x}\right)^2$

2) in deriving the Cauchy formula wrt. $z$ assuming the endstate were zero as well as the boundary but the initial state uniformly distributed, then there are 3 termd, the first involving second derivative of $n_\gamma(z)$ which gives a derivative of a delta and the integral gives a log type result. I don't know if this is correct ?

But assuming this would imply the square of the log is decreasing and then increasing, like a Bell signal.

But increasing signals with the distance is contrary to all forces in the nature ? Hence there shall be a maximal distance for intrication ?

 

[jk22]

Erratum : the imaginary part of theexpression in the integral is of the form $\frac{ct}{(x-s)^2+c^2t^2}ds$ so it is not a log but an arctan.

 

[jk22]

Addendum : if this is correct it could permit in particular to solve a lot of equations of the type $\box\psi=f(\psi)$ even with the requirement of compactness between an initial and and a final condition.

(Could The compactness solve for example the problem of infinite speed of diffusion of a delta localized particle ?)