Capacitance of horned torus

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Original poster: "Godfrey Loudner" <ggreen-at-gwtc-dot-net> 

Hello Antonio

A horned torus is a toroid with no hole. If one
tries to get the capacitance of a horned toroid
with toroidal coordinates, the essential calculations
collapse to 0 = 0. After looking around the internet,
I came across a list of strange coordinate system.
Tangent-sphere coordinates looked very interesting.
After trying the new coordinates, the solution to the
boundary value problem just did not look right.
After a long time, I finally realized that I was
missing a differential boundary condition (good
at math, but low on physical sense). With the
additional boundary condition, the rest of the
calculation was walk in the park. With d for the
inside diameter of a horned torus, the exact
expression for the capacitance is below.

Let a(n) be the ascending sequence of the positive zeros
of the zero order Bessel function J[0,x]. Then C =

8Pi(permittivity)d times the below

Sum[BesselJ[1,a(n)]^(-1)Integrate[E^(-a(n)Sinh[t]),{t,0,Infinity}],{n,1,Infi
nity}].

BesselJ[0,x] and BesselJ[1,x] are the zero and first order
Bessel function respectively.

The problem is computing the infinite series, but its a one
time calculation. My Mathematica can calculate the zeros of
Bessel J[0,x] and the integrals to 30 significant figures, but
I'm still having trouble getting the value. I have contacted
NASA, and they might run the calculation as a courtesy to my
school. I don't know yet if they will do the arithmetic. I've
asked them to calculate Pi times the infinite series to 20
significant figures.

Godfrey Loudner