if mode=TM
then Q:=QTM1
elif n=0
then Q:=QTM2
end if:

if mode=TE
then Q:=QTE1
elif m=0
then Q:=QTE2
elif n=0
then Q:=QTE3
end if:

This is the basic idea of what I'm looking for. 

However, for mode=TE, and m=0, n>0, I receive Q=QTE1, when I should have Q=QTE2.

Unfortunately it's by quite a few significant figures in some cases.

Also, after further investigation, the setup doesn't give correct values for each root.  Occasionally, the root corresponds to a value of n+1.

I also read that Maple has some trouble with interpreting the differential of the first kind of bessel function.

Thanks for the reply.

Unfortunately, that gives a different value than what I'm expecting for m=3, n=1.  I'd expect to get 4.2012 - http://mathworld.wolfram.com/BesselFunctionZeros.html towards the bottom of the page.

Edit: After looking at it, that value appears to be the root for m=3, n=2.  So, I changed the script to read:

f:=(m,n)->fsolve(BesselJ(m+1,x)*x-m*BesselJ(m,x),
    x=BesselJZeros(m,n-1)..BesselJZeros(m,n+1)):
f(3,1);
                             4.2012

However, for some other roots, it gives values which are not quite accurate.

Hi, thanks for all the responses.

I've found that Robert Israel's solution works out quite nicely, as I wanted to remain with the polar co-ords.

Thanks again.

Sorry for the expression - I had re-arranged it when I was using the fsolve command.

I have a circular geometry, examined in 2-D polar co-ordinates, r and phi.  The expression determines the magnitude of the electric field, at any given value of r and phi - i.e. at any point within the geometry.  What I want to do is try and create a contour plot of that field.

I had thought about using the solve command to solve for the radius, -radius..+radius, which would then leave the expression only in terms of phi.  Then, solve for phi, which would leave me with the magnitude of the field.

Is that a better explanation?