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Differential Equations by Pearson
book

Differential Equations by Pearson

by E. Rukmangadachari
May 2024
Intermediate to advanced content levelIntermediate to advanced
472 pages
13h 25m
English
Pearson India
Content preview from Differential Equations by Pearson
Integration in Series 6-17
6.1.8 Orthogonality of Functions
A set of functions
123
,,,ff f defi ned on some interval {|IxR=∈
}axb≤≤ is said to be orthogonal on I with respect to a
weight
function
() 0px >
if
() () () 0
b
mn
a
px f x f xdx=
for mn (6.33)
The norm
n
f
of
n
f
is defi ned by
2
() ()
b
nn
a
fpxfxdx=
(6.34)
The functions are called orthonormal on I if they are orthogonal on I
and all have a norm equal to 1.
In respect of functions with () 1,px = we simply say ‘orthogonal’.
Thus, functions
123
,,,ff f are orthogonal on some interval I if
() () 0
b
mn
a
fxfxdx=
for mn (6.35)
The norm
n
f
of
n
f
is then simply given by
2
()
b
nn
a
ffxdx=
(6.36) ...
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Publisher Resources

ISBN: 9781299487406Publisher Website