Math 621 Homework Assignment 2 Spring 2006
Due: Friday, February 24
1. (a) Let D1and D2be open connected sets in Cand f:D1→D2a holomorphic
map. Show that if His harmonic in D2, then the composition H◦fis
harmonic in D1.
(b) Let D⊂R2be a connected open set and u(x, y), v(x, y ) harmonic in D.
Prove or disprove the following statements:
i. The function w(x, y) = eu(x,y)is harmonic in D.
ii. The function w(x, y) = u(x, y)v(x, y ) is harmonic in D.
iii. If vis a harmonic conjugate of u, the function w(x, y ) = u2(x, y)−v2(x, y)
is harmonic in D.
2. Ahlfors page 28 problem 3: Find the most general harmonic homogeneous polyno-
mial of degree 3 (of the form ax3+bx2y+cxy2+dy3). Determine, by integration,
the conjugate harmonic function and the corresponding analytic function (up to a
constant).
3. Ahlfors page 28 problem 4: Show that if fis a holomorphic function with a
constant absolute value |f(z)|, then fitself is a constant function.
4. Lang page 58 problem 4a, c, d, g, h
5. Lang page 59 problem 10.
6. Ahlfors, page 41 problem 8: For what values of zis
∞
X
n=0 z
1 + zn
convergent?
(Describe the set geometrically).
7. Lang page 26 problem 7. Hint: Show first the following identity
zn−1
(1 −zn)(1 −zn+1)=1
(1 −z)2zn−1+zn−2+···+ 1
zn+zn−1+···+ 1 −zn−2+zn−3+· · · + 1
zn−1+zn−2+· · · + 1
8. Ahlfors, page 47 problem 8: Express arc tan(w) in terms of the logarithm.
9. Ahlfors, page 47 problem 9: Use an appropriate branch of log(z) to define the
angles of a triangle with vertices z1, z2, z3, bearing in mind that the angles should
be between 0 and π. With this definition, prove that the sum of the angles is π.
10. Lang Ch. II Sec 3 page 68 problem 4.
11. Find a fractional linear transformation that maps
(a) 0,1,∞to 1,−1,0,
(b) 0, i, −ito 1,−1,0.
12. Let T(z) = z−i
z+i. Determine the image of horizontal lines Im(z) = bunder T.
When the image is a circle, determine the center and radius (you may find it helpful
to use the notion of symmetry).
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