1000 Solved Problems in Modern Physics

(Romina) #1

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
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