Laplace's Equation--Bispherical Coordinates

In Bispherical Coordinates, Laplace's Equation becomes

$$\nabla^2f={\sin u\over(\cosh v-\cos u)^3}\left[{{\partial\ov... ...r\cosh v-\cos u}{\partial f\over\partial\phi}}\right)}\right].$$ (1)

Let

$$F(u,v,\phi)=\sqrt{\cosh v-\cos u}\,U(u)V(v)\Phi(\phi),$$

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

References

Arfken, G. ``Bispherical Coordinates $(\xi, \eta, \phi)$.'' §2.14 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 115-117, 1970.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 665-666, 1953.