Math 334
Assignment 8
Due: 12 Noon on Thursday, November 16, 2006.
1. (a) Evaluate I=R∞
0e−x2dx. (Hint: Consider I2and use double integration.)
(b) Show that Γ( 1
2) = √π. (Hint: Use the result of part (a).)
(c) Show that Γ(n+1
2) = (2n)!
n! 22n√π, for n∈N.
2. Show that Bessel functions of the first kind satisfy the following recurrence relations:
(a) d
dx (x−µJµ(x)) = −x−µJµ+1 (x);
(b) d
dx (xµJµ(x)) = xµJµ−1(x);
(c) 2µ
xJµ(x) = Jµ−1(x) + Jµ+1(x).
3. Show that
(a) J1
2(x) = r2
πx sin x; (b) J−1
2(x) = r2
πx cos x; (c) J3
2(x) = r2
πx sin x
x−cos x.
4. Show that Zℓ
0
xJ2
n(αnm
ℓx)dx =ℓ2
2J2
n+1(αnm), where αnm is the mth zero of Jn(i.e. Jn(αnm ) = 0).
Hint:
•Show that Bessel’s equation can be written as d
dx (xu′)2= (n2−x2)d
dx(u2), and integrate both
sides from x= 0 to an arbitrary point x=a.
•The formula obtained ab ove by integration must hold for all solutions of Bessel’s equation, so set
u=Jn.
•Use the formula from question 2a and set a=αnm to get the result.