The Chemistry Maths Book, Second Edition

15.2 Orthogonal expansions 419

Therefore

and this is in agreement with the coefficient ofP

0

shown in (15.6).

0 Exercises 1, 2

The general case

Letg

n

(x),n 1 = 1 1, 2, 3, =, be a set of functions, possibly complex, that are orthogonal

in the intervala 1 ≤ 1 x 1 ≤ 1 bwith respect to the weight functionw(x):

(15.10)

(g*

m

1 = 1 g

m

for real functions). Letf(x)be an arbitrary function, defined in the interval

a 1 ≤ 1 x 1 ≤ 1 b, that can be expanded in the set{g

n

(x)},

(15.11)

Then, multiplication byg*

m

(x)and integration with respect to weight functionw(x)

in the interval ato bgives

so that (replacing mby n),

(15.12)

The denominator in this expression is the normalization integral of the function

g

n

(x); it is the square of the norm

(15.13)

g

gxgx xdx

n

a

b

nn

= Z

*() () ()w

c

gxfx xdx

gxgx xdx

n

a

b

n

a

b

nn

=

Z

Z

*

() () ()

*() () ()

w

w

=cgxgxxdx

m

a

b

mm

Z

*

() () ()w

ZZ

a

b

m

n

n

a

b

mn

gxfx xdx c gxgx

*

()() ()

*

ww= () () (

=

∑

0

∞

xxdx)

fx cg x

n

nn

()= ()

=

∑

0

∞

Z

a

b

mn

gxgx xdx*() () ()w =≠0ifmn

ca

aaa a

l

l

l

00

246

0

2

357 21

=++++=

• 
  

=

∑



∞