Laplace's Equation--Toroidal Coordinates

In Toroidal Coordinates, attempt Separation of Variables to Laplace's Equation by plugging in the trial solution

$$f(u,v,\phi)=\sqrt{\cosh u-\cos v}\,U(u)V(v)\Psi(\psi),$$ (1)

then divide the result by $\mathop{\rm csch}\nolimits ^2 u(\cosh u-\cos v)^{5/2}$ $U(u)V(v)\Phi(\phi)$ to obtain

$${\textstyle{1\over 4}}\sinh^2 u+\cosh u\sinh u{U'(u)\over U(... ...}+\sinh^2 u{V''(v)\over V(v)}+{\Phi''(\phi)\over\Phi(\phi)}=0.$$ (2)

The function $\Phi(\phi)$ then separates with ${\Phi''(\phi)/\Phi(\phi)}=-m^2$, giving solution

$$\Psi(\psi)=\matrix{\sin\cr \cos\cr}(m\phi)=\sum_{k=1}^\infty [A_k\sin(m\psi)+B_k\cos(m\psi)].$$ (3)

Plugging $\Psi(\psi)$ back in and dividing by $\sinh^2 u$ gives

$$\coth u{U'(u)\over U(u)}+{U''(u)\over U(u)}-{m^2\over\sinh^2 u}+{1\over 4}+{V''(v)\over V(v)}=0.$$ (4)

The function $V(v)$ then separates with ${V''(v)/V(v)}=-n^2$, giving solution

$$V(v)=\matrix{\sin\cr \cos\cr}(nv)=\sum_{k=1}^\infty [C_k\sin(nv)+D_k\cos(nv)].$$ (5)

Plugging $V(v)$ back in and multiplying by $V(v)$ gives

$$U''(u)+\coth u U'(u)-\left[{{m^2\over\sinh^2 u}+(n^2-{\textstyle{1\over 4}})}\right]U(u)=0,$$ (6)

which can also be written

$${1\over\sinh u}{d\over du}\left({\sinh u{dU\over du}}\right)... ...t[{{m^2\over\sinh^2 u}+(n^2-{\textstyle{1\over 4}})}\right]U=0$$ (7)

(Arfken 1970, pp. 114-115). Laplace's Equation is partially separable, although the Helmholtz Differential Equation is not.

Laplace's Equation is partially separable, although the Helmholtz Differential Equation is not.

References

Arfken, G. ``Toroidal Coordinates $(\xi, \eta, \phi)$.'' §2.13 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 112-114, 1970.

Byerly, W. E. An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, p. 264, 1959.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 666, 1953.