Mathematical Methods for Physicists: A concise introduction - Site Map

SOLUTIONS OF LAPLACE'S EQUATION
In the special case k ˆ 0, Eq. (10.20) has solutions of the form
R…† ˆ d…† ‡ d 0 …† ; 6ˆ 0;
f ln ‡ g; ˆ 0:
…10:23†
When k 6ˆ 0, a simple change of variable can put Eq. (10.20) in the form of
Bessel's equation. Let x ˆ k, then dx ˆ kd and Eq. (10.20) becomes
d 2 R
dx 2 ‡ 1 dR
x dx ‡
1 2
x 2 !
R ˆ 0;
…10:24†
the well-known Bessel's equation (Eq. (7.71)). As shown in Chapter 7, R…x† can be
written as
R…x† ˆAJ …x†‡BJ …x†;
…10:25†
where A and B are constants, and J …x† is the Bessel function of the ®rst kind.
When is not an integer, J and J are independent. But when is an integer,
J …x† ˆ…1† n J …x†, thus J and J are linearly dependent, and Eq. (10.25)
cannot be a general solution. In this case the general solution is given by
R…x† ˆA 1 J …x†‡B 1 Y …x†;
…10:26†
where A 1 and B 2 are constants; Y …x† is the Bessel function of the second kind of
order or Neumann's function of order N …x†.
The general solution of Eq. (10.20) when k 6ˆ 0 is therefore
R…† ˆp…†J …k†‡q…†Y …k†;
…10:27†
where p and q are arbitrary functions of . Then these functions are also solutions:
H …1†
…k† ˆJ …k†‡iY …k†; H
…2† …k† ˆJ …k†iY …k†:
These are the Hankel functions of the ®rst and second kinds of order , respectively.
The functions J ; Y (or N ), and H
…1† ,andH
…2† which satisfy Eq. (10.20) are
known as cylindrical functions of integral order and are denoted by Z …k†,
which is not the same as Z…z†. The solution of Laplace's equation (10.15) can now
be written
8
ˆ…c 1 z ‡ b†… f ln ‡ g†; k ˆ 0; ˆ 0;
><
ˆ…c 1 z ‡ b†‰d…† ‡ d 0 …† Š‰a…†e i' ‡ a 0 …†e i' Š;
…; '; z†
k ˆ 0; 6ˆ 0;
ˆ‰c…k†e kz ‡ c 0 …k†e kz ŠZ 0 …k†; k 6ˆ 0; ˆ 0;
>:
ˆ‰c…k†e kz ‡ c 0 …k†e kz ŠZ …k†‰a…†e i' ‡ a 0 …†e i' Š; k 6ˆ 0; 6ˆ 0:
397