ABSTRACT
A good method for solving certain problems of the potential theory is to use the toroidal coordinates ϑ, ϕ, which are connected with the usual cylindrical coordinates r =
√ x2 + y2 and z by means of the formulae
r = sinhϑ
coshϑ− cosϕ ; z = sinϕ
coshϑ− cosϕ. (1)
If the problem to be solved shows axial symmetry, the potential Φ satisfying the Laplace equation in the physical space may be expressed, according to the formula
Φ = √
2(coshϑ− cosϕ)Ψ, (2)
by a function Ψ satisfying the equation
∂2Ψ ∂ϑ2
+ cothϑ ∂Ψ ∂ϑ
+ ∂2Ψ ∂ϕ2
= 0. (3)
This equation admits separation of variables. By putting
Ψ = p(ϑ)s(ϕ) (4)
one comes to the equations
d2p
dϑ2 + cothϑ
dp
dϑ + ( µ2 +
1 4
) p = 0, (5)
d2s
dϕ2 − µ2s = 0, (6)
where µ2 is a parameter. If, by the nature of the problem, the range of variation of ϑ is 0 ≤ ϑ < ∞, then the requirement that the solution be finite leads to positive values of the parameter µ2. In the case of eq. (5) the solution that meets this requirement is the Legendre function of the first kind with the complex index iµ− 12 and with the argument coshϑ
p(ϑ) = Piµ− 12 (coshϑ), (7)
while s(ϕ) can be put equal to
s(ϕ) = a(µ) coshµϕ + b(µ) sinhµϕ. (8)
From the particular solution of shape (4) a more general solution may be derived:
Ψ = ∫ ∞ 0
Piµ− 12 (coshϑ){a(µ) coshµϕ + b(µ) sinhµϕ}dµ. (9)
The problem arises to determine the functions a(µ) and b(µ) from some boundary conditions for Ψ. When these conditions include the function ψ or a linear combination of Ψ and ∂Ψ∂ϕ is given for two values of the coordinate ϕ, the problem evidently reduces to the determination of the function f(µ) from a given function ψ(µ), so as to have
ψ(x) = ∫ ∞ 0
Piµ− 12 (x)f(x)dµ (1 ≤ x <∞). (10)
In other words, the problem is reduced to the inversion of the integral (10) and to the expansion of an arbitrary function by the Legendre functions with a complex index. It is just this problem that we are to consider in the present article.
