632 Chapter^8
= (a)m+l + ~ (7)(a)m-k+1(b)k
- ~ (k: 1 )(a)m-k+t(b)k + (b)m+t
= (a)m+i + L m ( m k +1) (a)m+t-k(b)k + (b)m+t
k=l
m+l ( + l)
= ~ m k (a)m+i-k(b)kwhich shows that the formula: also holds for n = m + 1. Hence it is valid
for an arbitrary positive integer n.
Exercises 8.9
- Use the f-function to evaluate the following integrals.
(a) lCXJ e-x
4
dx (b) 1CXJ xe-x3
dx(c) :lCXJ e-a2x2 dx (a> 0) o(d) 1
1xmln(~)n dx (m > -1,n > 0)
(e) 1CXJ e_xs1;-dx (f) 11 c:x r/3 dx
- Show that:
(a) [CXJ xne-a2x2 dx = r((n + 1)/2) (a> 0, n > -1)
h ~n+l
[CXJ xa r(a + 1)
(b) Jo ax dx = (lna)a+l (a > 1)
(c) r(2n+l) = 1·3·5···(2n-l)Vir= (2n-l)!!v;;rt
2 2n 2n- Prove that I'(z + 1) = zr(z) by using Euler's formula (8.20-19).
- Prove that
B(a, b)I'(a + b + n) = t (~)r(a + n - k)I'(b + k)
k=OtThe notation (2n -1 )!! = 1 · 3 · 5 · · · (2n -1) is called a semifactorfol. Similarly,
(2n)!! = 2 · 4 · 6···(2n).