Math 535 Homework #4
Winter 2008
In all of the problems b elow, be sure you have defined your functions carefully. Prove all claims
about integrals. Draw a picture of any contour you use.
1. For 0 < α < 1, find
Z∞
0
xα
x(x+ 1) dx.
2. Find the inverse Laplace transform of
F(z) = 3z2+ 12z+ 8
(z+ 2)2(z+ 4)(z−1)
using the residue theorem.
3. Compute
ζ(6) =
∞
X
n=1
1
n6
by calculus of residues. You can use your answer to problem 3(e) on HW 3. (Remark: This process
can be used to compute ζ(2n). The value of ζat an odd integer is another story.)
4. Verify:
∞
X
n=0
(−1)n
(2n+ 1)3=π3
32 .
Hint: What are the residues of π / sin πz?
5. Find: Z∞
0
x3+ 8
x5+ 1 dx,
using a contour integral of (log z)(z3+ 8)/(z5+ 1).
6. Find
Z1
0
1
px(1 −x)dx.
Put a “dog bone” around the interval [0,1] and add a large circle. Carefully define the integrand
so that it is analytic at ∞, then the integral over the large circle can be found from the series
expansion at ∞.Redo the problem by first making the substitution x= 1/w.