Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SPECIAL FUNCTIONS

Show that theth spherical Bessel function is given by

f(x)=(−1)x

(

1

x

d

dx

)

f 0 (x), (18.106)

wheref(x)denotes eitherj(x)orn(x).

The recurrence relation (18.93) for Bessel functions of the first kind reads

Jν+1(x)=−xν

d

dx

[

x−νJν(x)

]

.

Thus, on settingν=+^12 and rearranging, we find

x−^1 /^2 J+3/ 2 (x)=−x

d

dx

[

x−^1 /^2 J+1/ 2

x

]

,

which on using (18.102) yields the recurrence relation

j+1(x)=−x

d

dx

[x−j(x)].

We now change+1→and iterate this result:

j(x)=−x−^1

d

dx

[x−+1j− 1 (x)]

=−x−^1

d

dx

{

x−+1(−1)x−^2

d

dx

[

x−+2j− 2 (x)

]

}

=(−1)^2

x

x

d

dx

{

1

x

d

dx

[

x−+2j− 2 (x)

]

}

=···

=(−1)x

(

1

x

d

dx

)

j 0 (x).

This is the expression forj(x) as given in (18.106). One may prove the result (18.106) for
n(x) in an analogous manner by settingν=−^12 in the recurrence relation (18.92) for
Bessel functions of the first kind and using the relationshipY+1/ 2 (x)=(−1)+1J−− 1 / 2 (x).

Using result (18.106) and the expressions (18.104) and (18.105), one quickly

finds, for example,

j 1 (x)=

sinx

x^2

−

cosx

x

, j 2 (x)=

(

3

x^3

−

1

x

)

sinx−

3cosx

x^2

,

n 1 (x)=−

cosx

x^2

−

sinx

x

, n 2 (x)=−

(

3

x^3

−

1

x

)

cosx−

3sinx

x^2

.

Finally, we note that the orthogonality properties of the spherical Bessel functions

follow directly from the orthogonality condition (18.88) for Bessel functions of

the first kind.

18.7 Laguerre functions

Laguerre’s equation has the form

xy′′+(1−x)y′+νy= 0; (18.107)