1000 Solved Problems in Modern Physics

30 1 Mathematical Physics

1.76 The Bessel functionJn(x) is given by the series expansion

Jn(x)=

∑

(−1)k(x/2)n+^2 k

k!Γ(n+k+1)

Show that:

(a)ddx[xnJn(x)]=xnJn− 1 (x)

(b)ddx[x−nJn(x)]=−x−nJn+ 1 (x)

1.77 Prove the following relations for the Bessel functions:

(a)Jn− 1 (x)−Jn+ 1 (x)= (^2) ddxJn(x)
(b)Jn− 1 (x)+Jn+ 1 (x)= 2 nxJn(x)
1.78 Given thatΓ

( 1

2

)

=

√

π, obtain the formulae:

(a)J 1 / 2 (x)=

√

2

πxsinx

(b)J− 1 / 2 (x)=

√

2

πxcosx

1.79 Show that the Legendre polynomials have the property:

∫l

−l

Pn(x)Pm(x)dx=

2

2 n+ 1

,ifm=n

= 0 ,ifm

=n

1.80 Show that for largenand smallθ,Pn(cosθ)≈J 0 (nθ)

1.81 For Legendre polynomialsPl(x) the generating function is given by:

T(x,s)=(1− 2 sx+s^2 )−^1 /^2

=

∑∞

l= 0

Pl(x)sl,s< 1

Use the generating function to show:

(a) (l+1)Pl+ 1 =(2l+1)xPl−lPl− 1

(b)Pl(x)+ 2 xPl′(x)=Pl′+ 1 (x)+Pl′− 1 (x), Where prime means differentiation

with respect tox.

1.82 For Laguerre’s polynomials, show thatLn(0)=n!. Assume the generating

function:

e−xs/(1−s)

1 −s

=

∑∞

n= 0

Ln(x)sn

n!

1.2.11 ComplexVariables.................................

1.83 Evaluate

∮

c

dz

z− 2 whereCis:

(a) The circle|z|= 1

(b) The circle|z+i|= 3