Mathematical Methods for Physicists: A concise introduction - Site Map

SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS
From Eq. (7.76a) we obtain J 0
0 as
J 0
0…x† ˆX1
mˆ1
By inserting this series we have
X 1
mˆ1
…1† m 2mx 2m1
2 2m …m!† 2 ˆ X1
…1† m x 2m1
2 2m2 m!…m 1†! ‡ X1
mˆ1
mˆ1
m 2 A m x m1 ‡ X1
…1† m x 2m1
2 2m1 m!…m 1†! :
mˆ1
A m x m‡1 ˆ 0:
We ®rst show that A m with odd subscripts are all zero. The coecient of the
power x 0 is A 1 and so A 1 ˆ 0. By equating the sum of the coecients of the power
x 2s to zero we obtain
…2s ‡ 1† 2 A 2s‡1 ‡ A 2s1 ˆ 0; s ˆ 1; 2; ...:
Since A 1 ˆ 0, we thus obtain A 3 ˆ 0; A 5 ˆ 0; ...; successively. We now equate the
sum of the coecients of x 2s‡1 to zero. For s ˆ 0 this gives
For the other values of s we obtain
For s ˆ 1 this yields
1 ‡ 4A 2 ˆ 0 or A 2 ˆ 1=4:
…1† s‡1
2 s …s ‡ 1†!s! ‡…2s ‡ 2†2 A 2s‡2 ‡ A 2s ˆ 0:
1=8 ‡ 16A 4 ‡ A 2 ˆ 0 or A 4 ˆ3=128
and in general
A 2m ˆ …1†m1
2 m …m!† 2 1 ‡ 1 2 ‡ 1 3 ‡‡ 1
;
m
m ˆ 1; 2; ...: …7:82†
Using the short notation
h m ˆ 1 ‡ 1 2 ‡ 1 3 ‡‡ 1 m
and inserting Eq. (7.82) and A 1 ˆ A 3 ˆˆ0 into Eq. (7.81) we obtain the result
y 2 …x† ˆJ 0 …x† ln x ‡ X1
mˆ1
…1† m1 h m
2 2m …m!† 2 x2m
ˆ J 0 …x† ln x ‡ 1 4 x2 3
128 x4 ‡: …7:83†
Since J 0 and y 2 are linearly independent functions, they form a fundamental
system of Eq. (7.80). Of course, another fundamental system is obtained by
replacing y 2 by an independent particular solution of the form a…y 2 ‡ bJ 0 †,
where a…6ˆ 0† and b are constants. It is customary to choose a ˆ 2= and
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