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562 CHAPTER 15 Special Functions and Eigenfunction Expansions

the nth positive zero of J 1 . Compare a graph of this partial 37. Show that, if r > 0, then for any positive x,
sum with f . ∞
t

(x) =r x e −rt x−1 dt.
24. f (x) = e −x 0
25. f (x) = x Hint:Let t =ry in the definition of
(x).
2 −x
26. f (x) = x e 38. Show that, for positive x,
27. f (x) = xe −x ∞

(x) = 2 e −t 2 2x−1 dt.
t
28. f (x) = x cos(πx)
0
29. f (x) = sin(πx)
2
Hint:Let t = y in the definition of the gamma
function.
For each of Problems 30 through 35, find (approximately)
39. Define the beta function by
the first five terms in the Fourier-Bessel expansion of f (x)
on (0,1) in a series of the functions J 2 ( j n x), with j n the 1
B(x, y) = t x−1 (1 − t) y−1 dt.
nth positive zero of J 2 .
0
It can be shown that this integral converges for x and
30. f (x) = e −x
y positive. Show that
31. f (x) = x
∞ x−1
2 −x
32. f (x) = x e u
B(x, y) = du.
33. f (x) = xe −x 0 (1 + u) x+y
34. f (x) = x cos(πx) Hint:Let t = u/(1 + u) in the definition of B(x, y).
35. f (x) = sin(πx) 40. Show that, if x and y are positive integers, then

(x)
(y)
Problems 36 through 40 deal with the gamma and beta B(x, y) = .

(x + y)
functions.
x
(x+1)
Hint: Begin with L[t ]= s k+1 . Now compute
√
1
36. Show that
(1/2) = π, using the fact from statistics L −1 s x+y in two ways, first by using this formula,
∞ −x 2 √
that e dx = π/2. and second, by using the convolution theorem.
0

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October 14, 2010 15:20 THM/NEIL Page-562 27410_15_ch15_p505-562