Mathematical Methods for Physicists: A concise introduction - Site Map

SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS
7.10 Verify Eq. (7.37) for the function P 1 2…x†:
7.11 Show that
d nm
dx nm …x2 1† n …n m†! ˆ
…n ‡ m†! …x2 1† m d n‡m
dx n‡m …x2 1† m
Hint: Write …x 2 1† n ˆ…x 1† n …x ‡ 1† n and ®nd the derivatives by
Leibnitz's rule.
7.12 Use the generating function for the Hermite polynomials to ®nd:
(a) H 0 …x†; (b) H 1 …x†; (c) H 2 …x†; (d)H 3 …x†.
7.13 Verify that the generating function satis®es the identity
@ 2
@x
@
2x
2
@x
‡ 2t
@
@t ˆ 0:
Show that the functions H n …x† in Eq. (7.47) satisfy Eq. (7.38).
7.14 Given the di€erential equation y 00 ‡…" x 2 †y ˆ 0, ®nd the possible values
of " (eigenvalues) such that the solution y…x† of the given di€erential equation
tends to zero as x !1. For these values of ", ®nd the eigenfunctions
y…x†.
7.15 In Eq. (7.58), write the series for the exponential and collect powers of z to
verify the ®rst few terms of the series. Verify the identity
x @2 @
‡…1 x†
@x2 @x ‡ z @
@z ˆ 0:
Substituting the series (7.58) into this identity, show that the functions L n …x†
in Eq. (7.58) satisfy Laguerre's equation.
7.16 Show that
7.17 Show that
J 0 …x† ˆ1
x2
2 2 …1!† 2 ‡ x4
2 4 …2!† 2 x6
2 6 …3!† 2 ‡;
J 1 …x† ˆx
2 x3
2 3 1!2! ‡ x5
2 5 2!3! x7
2 7 3!4! ‡:
J 1=2 …x† ˆ 2 1=2
sin x; J
x
1=2 …x† ˆ 2 1=2
cos x:
x
7.18 If n is a positive integer, show that the formal expression for J n …x† gives
J n …x† ˆ…1† n J n …x†.
7.19 Find the general solution to the modi®ed Bessel's equation
x 2 y 00 ‡ xy 0 ‡…x 2 s 2 2 †y ˆ 0
which di€ers from Bessel's equation only in that sx takes the place of x.
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