A treatise on the theory of Bessel functions, by G. N. Watson.

124

THEORY OF BESSEL FUNCTIONS

[CHAP. IV

4*8. Solutions of Laplace's equation.
The first appearance in analysis of the general Bessel coefficient has been
seen (~ 1'3) to be in connexion with an equation, equivalent to Laplace's
equation, which occurs in the problem of the vibrations of a circular membrane.
We shall now shew how Bessel coefficients arise in a natural manner from
Whittaker's* solution of Laplace's equation
a2V a y ayV
(1) t 2 +y2 O.
axa +~+ =2 o
The solution in question is
(2) V = f(z + ix cos u + iy sin u, u) du,
-7r
in which f denotes an arbitrary function of the two variables involved.
In particular, a solution is
ek (z+ixcosu+iysinu) COS mu dC,
- 7
in which k is any constant and m is any integer.
If we take cylindrical-polar coordinates, defined by the equations
x=pcosO, y=psin b,
this solution becomes
ekz eikpCos(u-4) cos mtdu = ekz eikpcosv cos m (v + b) dv,
v - T r -V
r.
= 2ekz eikp00sV cos my cos m nA dv,
J 0
= 2rri ekZ cos mQ,. Jm (kp),
by ~ 2-2. In like manner a solution is
ek (z + ix cos u + iy sin u) sin mu du,
—,
and this is equal to 27rii ek sin mob. Jm (kp). Both of these solutions are
analytic near the origin.
Again, if Laplace's equation be transformedt to cylindrical-polar coordinates, it is found to become
a2V IJV 1 a2V a2V
+- + + ~ =;
ap2 p ap p2 aq2 +2=
* Monthly Notices of the R. A. S. LXII. (1902), pp. 617-620; Math. Ann. LVII. (1902),
pp. 333-341.
t The simplest method of effecting the transformation is by using Green's theorem. See
W. Thomson, Camb. Math. Journal, iv. (1845), pp. 33-42.