Error executing triple intergration equation. I am getting 'Not a Number (NaN)' as output for the triple integration equation attached below (Image file). I have mentioned the code used also. Kindly help me in knowing where I am wrong.

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syms x
syms y
syms phi
fun=@(x,y,phi) (cos(phi).*(x.*y))./(sqrt((x.^2)+(y.^2)-(2.*x.*y.*cos(phi))))
q = NaN
L = NaN
David Goodmanson
David Goodmanson on 15 Apr 2019
Edited: David Goodmanson on 15 Apr 2019
Hi Torsten,
Not totally obvious, but if xvec and yvec are vectors expressed in polar coordinates (x,phix) and (y,phi) about the origin, this starts out as the integral
cos(phi-phix) / |yvec-xvec|
with area elements
xdx dphix and ydy dphi,
integrated over a circular annulus of inner and outer radii a and b. Toss in some constants and it's the self inductance of the annulus.
After doing the y and phi integrations, by symmetry the answer can't depend on phix. So you can assume that phix = 0 and multiply by 2pi later. This gives the integral in the original question, which has a singularity at phi=0, x=y as you mentioned. But cos(phi) = 1 there so the integrand is basically 1/|yvec-xvec|. Take xvec as the new origin and coordinates r and theta to describe yvec-xvec. For a small circle of radius B centered about xvec, this becomes
Integral (1/r) rdr dtheta = 2pi B
so the integral is bounded. But it takes a new set of coordinates to allow the new area element to cancel out the 1/r behavior. That's why I mentioned separate regions of integration before.

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