The Chemistry Maths Book, Second Edition

386 Chapter 13Second-order differential equations. Some special functions

The solutions of the indicial equation (13.8) are r 1 = 1 ±n, and the two particular

solutions for these values of rare

(13.48)

(13.49)

Important values of the parameter nare the integer and half-integer values.

Bessel functions J

n

(x)for integer n

When nis a positive integer or zero, the particular solution (13.48) is

(13.50)

in which the constant a

0

has been given its conventional value of a

0

1 = 112 (2

n

1 n!).

This is the Bessel function of the first kind of order n. The function is finite and

converges for all values of x; it converges very fast because of the pair of factorials

in the denominator. The functions forn 1 = 10 andn 1 = 11 are

(13.51)

(13.52)

and their graphs are shown in Figure 13.2.

Jx

xx x

1

35

2

1

12 2

1

23 2

1

34

()=−

!!













!!













−

!!!













7

2

x

Jx

xx

0 2

2

4

1

2

1

2

1

3

()

() ( ) (

=−

!











 +

!











 −

!!











 +

)

2

6

2

x

Jx

x

mn m

x

n

m

()

=













−

!+!













=

∑

2

1

2

0

2

∞

yx ax

x

n

x

nn

n

20

24

1

22 2 2 42 2 2 4

()

() ()()

=+

−

−−



−

·











yx ax

x

n

x

nn

n

10

24

1

22 2 2 42 2 2 4

()

()()()

=−

++

−





·











10

0

x

1

J 0

J 1

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Figure 13.2