Math 621 Homework Assignment 8 Spring 2006
Due: Wednessday, May 24
1. Compute the following integrals:
(a) Zπ/2
0
dx
a+ sin2(x),|a|>1,
(b) Z∞
0
x2dx
x4+ 5x2+ 6,
(c) Z∞
0
cos(x)
x2+a2dx, a > 0 real,
(d) Z∞
0
xsin(x)
x2+a2dx, a ≥0,real,
(e) Z∞
0
x1/3
1 + x2dx
2. (a) Basic Exam September 98 Problem 9b: Prove that the image of the complex
plane C, under a non-constant entire function, is dense in C.
(b) Prove that there does not exist a one-to-one conformal map from the complex
plane Conto the unit disk.
3. Lang page 307 Problem 7: The holomorphic automorphism group of a simply
connected open set Uacts transitively on points. More precisely, let U⊂Cbe
a simply connected open set, z1,z2two points in U. Use the Riemann-Mapping-
Theorem (Lang, page 306) to prove that there exists a holomorphic automorphism
fof Usuch that f(z1) = z2.Distinguish the cases when U=Cand U6=C.
4. Basic Exam, January 99 Problem 7: Find a one-to-one conformal map from the
region obtained as the intersection of the two unit disks:
{z:|z|<1} ∩ {z:|z−1|<1}
onto the upper half plane. Hint: show that the angle between the two circles, at
each of the two points of intersection, is 2π/3. For the smoothing of the boundary,
observe that the function z1/α,0< α ≤2, is wel l defined on the region Ω := {z:
Arg(z)∈(0, α ·π)}(why?) and maps Ωonto the upper half plane.
Harmonic functions and boundary value problems: Aboundary value prob-
lem, or Dirichlet problem is the problem of finding a harmonic function with given
boundary values. For example, the function arg(z), with values in [0, π], is the
unique harmonic function hin the upper-half-plane, satisfying the boundary prob-
lem h(x, 0) = 0 if x > 0
πif x < 0(and with a jump singularity at the boundary point
(0,0)). There is a nice integral formula (the Poisson formula), for recovering the
harmonic function given its boudary values (Ch 6 is Ahlforse, Ch VIII sec 4 in
Lang). The problems below can all be solved without it, using the techniques of
fractional linear transformations studied in this course, combined with the exam-
ples of harmonic functions we encountered as real and imaginary parts of elemen-
tary holomorphic functions.
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