The Chemistry Maths Book, Second Edition

422 Chapter 15Orthogonal expansions. Fourier analysis

EXAMPLE 15.2Derive the expansion (15.22).

We treat the function to be expanded as a function of t, and expand it as a MacLaurin

series. We have

f(t) = 1 (1 1 − 12 xt 1 + 1 t

2

)

− 122

f(0) = 11

f′(t)= 1 (x 1 − 1 t) (1 1 − 12 xt 1 + 1 t

2

)

− 322

f′(0)= 1 x

f′′(t) 1 = 1 3(x 1 − 1 t)

2

(1 1 − 12 xt 1 + 1 t

2

)

− 522

1 − 1 (1 1 − 12 xt 1 + 1 t

2

)

− 322

f′′(0) 1 = 13 x

2

1 − 11

and similarly,f′′′(0) 1 = 115 x

3

1 − 19 x,f′′′′(0) 1 = 1105 x

4

1 − 190 x

2

1 + 19 , and so on.

Comparison with the Legendre polynomials listed in (15.4) shows that, for the lth

derivative,f

(l)

(0) 1 = 1 l!P

l

(x). The MacLaurin series

f′′′′(0) 1 +1-

is then

f(t) 1 = 1 (1 1 − 12 xt 1 + 1 t

2

)

− 122

1 = 1 P

0

1 + 1 tP

1

(x) 1 + 1 t

2

P

2

(x) 1 + 1 t

3

P

3

(x) 1 + 1 t

4

P

4

(x) 1 +1-

and this is equation (15.22).

EXAMPLE 15.3Find the first term in the expansion (15.23).

If then. Therefore

It follows from equations (13.55) and (13.59) for Bessel functions of half-integral

order that

0 Exercises 3, 4

The expansion of electrostatic potential

The potential of a single charge

The electrostatic potential at point P due to the presence of a

point charge qis (see Figure 15.1)

V (15.24)

q

R

q

=

4

0

πε

c

t

Jtjt

0120

2

==

/

π

() ()

cedx

e

it i

itx

itx

0

1

1

1

1

1

2

1

2

1

2

==





















=

−

+

−

+

Z

tt

ee

t

t

it it

−













=

−

−

+

1

1

1

sin

c

l

Pxe dx

ll

itx

=

• 
  

−

+

21

2

1

1

Z ()

ecPx

itx

l

ll

=

=

∑

0

∞

()

ft f tf

t

f

t

f

t

() () ()=+′ + () ()

!

′′ +

!

00 ′′′ +

2

0

3

0

4

23 4

!!

.

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• 
  
  
  
  • 
    
    
    
    • 
      
    
    
    
    
  
  
  
  

o

q

p

r

R

R

q

..

...

...

...

....

....

.....

θ

Figure 15.1