Mathematical Methods for Physicists: A concise introduction - Site Map

FOURIER SERIES AND INTEGRALS
By the boundary condition u…l† ˆ0 we ®nd Ce kl ‡ De kl ˆ 0, and G > can be
rewritten as
G > ˆ C 0 ‰e k…xl† e k…xl† Š;
…4:67b†
where C 0 ˆ Ce kl .
How do we determine the constants A and C 0 ? First, continuity of G at x ˆ x 0
gives
A…e kx e kx †ˆC 0 …e k…xl† e k…xl† †:
…4:68†
A second constraint is obtained by integrating Eq. (4.61) from x 0 " to x 0 ‡ ",
where " is in®nitesimal:
Z " #
x 0 ‡"
d 2 Z
G
x 0 ‡"
dx 2 k2 G dx ˆ …x x 0 †dx ˆ 1: …4:69†
But
x 0 "
Z x 0 ‡"
x 0 "
x 0 "
k 2 Gdx ˆ k 2 …G > G < †ˆ0;
where the last step is required by the continuity of G. Accordingly, Eq. (4.64)
reduces to
Z x 0 ‡"
x 0 "
d 2 G
dx 2 dx ˆ dG >
dx dG <
dx ˆ 1:
…4:70†
Now
dG <
dx xˆx 0
ˆ Ak…e kx 0 ‡ e kx 0 †
and
dG >
dx xˆx 0
ˆ C 0 k‰e k…x 0 l† ‡ e k…x 0 l† Š:
Substituting these into Eq. (4.70) yields
C 0 k…e k…x 0 l† ‡ e k…x 0 l† †Ak…e kx 0 ‡ e kx 0 †ˆ1:
…4:71†
We can solve Eqs. (4.68) and (4.71) for the constants A and C 0 . After some
algebraic manipulation, the solution is
A ˆ 1
2k
sinh k…x 0 l†
; C 0 ˆ 1 sinh kx 0
sinh kl
2k sinh kl
194