MATH 440/508.101 - FINAL EXAM
Due on Friday December 10th, 2010 at 1pm in MATH219
Name —————————————–
Student number —————————
Problem 1:
Use Residue calculus to prove that
(a)
Z
∞
sin(x2 ) dx =
√
2π/4.
0
Z
∞
2
e−x dx =
Hint: You may use the equality
√
π.
−∞
(b)
Z
1
log(sin πx) dx = − log 2.
0
Problem 2:
Let d(n) denote the number of divisors of n and
f (z) =
∞
X
d(n)z n
for |z| < 1.
n=1
(a) Prove that f is analytic on the unit disk.
(b) Verify the identity
∞
X
n=1
n
d(n)z =
∞
X
n=1
zn
.
1 − zn
(c) Prove that for r ∈ (0, 1) and p, q ∈ N, we have
|f (rei2πp/q )| ≥ cp/q
1
log(1/(1 − r))
1−r
as r → 1.
(d) Conclude that f can not be extended analytically past the unit disk.
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Problem 3:
Assume that ζ(s) is the Riemann zeta function and Q(x) = {x} − 1/2 where {x} is the
fractional part of x.
(a)Prove that
1
s
− −s
ζ(s) =
s−1 2
Z
1
∞
Q(x)
d x.
xs+1
(b) Assume Qk (x) is defined recursively so that
Z
1
Qk (x) d x = 0,
0
dQk+1 (x)
= Qk (x),
dx
Prove that
s
1
ζ(s) =
− −s
s−1 2
Q0 (x) = Q(x) and Qk (x + 1) = Qk (x).
Z
∞
1
dk
Qk (x) x−s−1 d x.
dxk
(c) Use this formula to define an analytic extension for ζ(s) when Re(s) > −k.
(d) Prove that Qk (x) =
1
B (x)
(k+1)! k+1
where Bk+1 (x) are the Bernoulli polynomials.
(e) Prove that for k positive integer we have
2ζ(2k) = (−1)k+1
where B2k = B2k (0) are the Bernoulli numbers.
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(2π)2k
B2k ,
(2k)!
Problem 4:
Suppose f is an injective analytic function on the unit disk D, such that f (0) = 0 and
∞
X
f 0 (0) = 1. Write f (z) =
an z n . Assume an ∈ R for all n.
n=1
(a) Let z = reiθ with r ∈ (0, 1), and v(r, θ) denote the imaginary part of f (reiθ ), prove that
Z
2 π
n
v(r, θ) sin nθ dθ.
an r =
π 0
(b) Show that for 0 ≤ θ ≤ π and n ∈ N we have | sin nθ| ≤ n sin θ.
(c) Prove that f (D) is symmetric with respect to the real axis, and use this fact to show
that f maps the upper half disk into either the upper or lower part of f (D).
(d) Prove that for r small,
v(r, θ) = r sin θ [1 + O(r)],
and use the previous part to conclude that v(r, θ) sin θ ≥ 0 for all r ∈ (0, 1) and θ ∈ [0, π].
(e) Prove that for all integers n and all r ∈ (0, 1) we have |an rn | ≤ nr, and therefore |an | ≤ n.
(f) Check that function f (z) = z/(1 − z)2 satisfies all the hypotheses and |an | = n for all n.
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Problem 5:
Assume h(x) is a continuous function supported on [−M, M ] and let
1
g(z) =
2πi
Z
+M
−M
h(x)
dx.
x−z
(a) Prove that g is analytic on C\[−M, M ], and vanishes at infinity. Moreover, prove that
the jump of g across [−M, M ] is h, that is:
h(x) = lim+ g(x + i) − g(x − i).
→0
(b) Assume h satisfies a Hölder condition with positive exponent α, that is |h(x) − h(y)| ≤
c |x − y|α for some constant c > 0 and all x, y ∈ [−M, M ]. Prove that g(x + i) and g(x − i)
converge uniformly as → 0.
(c) Prove g can be characterized as the unique function which satisfies:
i. g is an analytic function on C\[−M, M ],
ii. g vanishes at infinity,
iii. g(x + i) and g(x − i) converge uniformly as → 0 to functions g+ (x) and g− (x) with
g+ (x) − g− (x) = h(x).
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