I have an issue when i try to follow the following reasoning in Snieder book in a cylindrical coordinate system. To summarize the reasoning:
consider a harmonic function V(x,z) so that its Laplacian is zero $\Delta V(x,z)=0$
consider its Fourier transform over $x$, $v(k,z)$
\begin{array}{l} V(x,z) = \int\limits_\infty {v(k,z){e^{ikx}}dk} \end{array}
- expressing the cylindrical Laplacian (i must do it like that because my function of interest yields zero Laplacian using the cylindrical expression for the $x$ variable - my system is a bit unusual as i have to consider the $x$ variable to be cylindrical even though i work in a $xz$ plane, no angle involved)
EDIT: example: the gravity field profile of a vertical cylinder at depth $z$ is expressed as $V(x,z)=\frac{1}{{\sqrt {{x^2} + {z^2}} }}$, with $x$ in $(-\infty,+\infty)$. Only the cylindrical Laplacian of this function is zero, not the cartesian.
\begin{array}{l} \Delta V(x,z) = \int\limits_\infty {\left( {\frac{1}{x}\frac{\partial }{{\partial x}}\left( {x\frac{\partial }{{\partial x}}} \right) + \frac{{{\partial ^2}}}{{\partial {z^2}}}} \right)\left[ {v(k,z){e^{ikx}}} \right]dk} \\ = \int\limits_\infty {\left( {\frac{1}{x}\frac{\partial }{{\partial x}}\left( {xikv(k,z){e^{ikx}}} \right) + \frac{{{\partial ^2}}}{{\partial {z^2}}}(v(k,z){e^{ikx}})} \right)dk} \\ = \int\limits_\infty {\left( {\frac{1}{x}ikv(k,z)\frac{\partial }{{\partial x}}\left( {x{e^{ikx}}} \right) + \frac{{{\partial ^2}}}{{\partial {z^2}}}(v(k,z){e^{ikx}})} \right)dk} \\ = \int\limits_\infty {\left( {\frac{{ik}}{x}v(k,z)\left( {{e^{ikx}} + xik{e^{ikx}}} \right) + \frac{{{\partial ^2}v(k,z)}}{{\partial {z^2}}}{e^{ikx}}} \right)dk} \\ = \int\limits_\infty {\left( {\frac{{ik(1 + ikx)}}{x}v(k,z) + \frac{{{\partial ^2}v(k,z)}}{{\partial {z^2}}}} \right){e^{ikx}}dk} = 0 \end{array}
- The equation being equal to zero, and the term between brackets being the Fourier coefficients of the $\Delta V$, the Fourier coefficients can be equalled to zero, which yields the following PDE :
$\frac{{{\partial ^2}v(k,z)}}{{\partial {z^2}}} - (k^2-\frac{{ik}}{x})v(k,z) = 0$
- To me this yields the following type of solution:
$v(k,z) = A(k)\exp \left( { + \sqrt {\left| {{k^2} - i\frac{k}{x}} \right|} .z} \right) + B(k)\exp \left( { - \sqrt {\left| {{k^2} - i\frac{k}{x}} \right|} .z} \right)$
I have some problems with this:
$v(k,z)$ is not supposed to have any $x$ variable in it, because it is the Fourier transform of $V(x,z)$ along $x$
Is there a fundamental error here? Or does it just mean that the general solutions do not apply in my case? (I'm not sure if that is even possible?)
I just don't know what to make of this situation. If someone sees a mistake in what I wrote, or can interpret what the apparent contradiction means, please point it out.