and when pn= is an integer the general solution is
1212
()()()()
nnnn
yCJ tCYtCJxCYx
λλ
=+=+
6.2.12 Orthogonality
Defi nition of orthogonality of functions
A set of functions
12
,,ff…
defi ned on some interval [,,]Iab= is said
to be ortho
gonal on I with respect to a weight function
()0wx>
if
()()()0
b
mn
a
wxfxfxdx=
∫
for mn≠
The norm
m
f
of
m
f
is defi ned by
2
()()
b
mm
a
fwxfxdx=
∫
The functions are called orthonormal of I if they are orthogonal
on I and all have norm equal to unity.
For ‘orthogonal w.r.t. w(x) = 1’ we simply say orthogonal
on I. Thus, functions
12
,,ff… are orthogonal on some interval I if
()() ...
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