Math 621 Homework Assignment 1 Spring 2006
Due: Friday, February 10
1. Lang, Ch. 1 Section 2 page 11: 7, 10, 11, 12, 13
2. Lang, Ch. 1 Section 3 page 17 problem 4: Let f(z) = ez. Describe the image
under fof the following sets:
(a) {z=x+iy |x≤1 and 0 ≤y≤π}.
(b) {z=x+iy |0≤y≤π(no condition on x)}.
3. Lang, Ch. 1 Section 4 page 26 problem 3: Show that for any complex number
z6= 1, we have
1 + z+···+zn=zn+1 −1
z−1.
If |z|<1, show that
lim
n→∞(1 + z+···+zn) = 1
1−z
4. Let u,vbe real valued functions defined on an open set Uin R2. Prove that if u
and vhave continuous partial derivatives, then the function
f:U→R2
(x, y)7→ (u(x, y), v (x, y))
is differentiable throughout Uin the sense that
lim
(∆x,∆y)7→(0,0)
kf(x+ ∆x, y + ∆y)−[f(x, y) + df(∆x, ∆y)] k
k(∆x, ∆y)k= 0.
Hint: Bound the quotient by a sum of two terms each depending only on uor v,
define a(t) := u(z+t·∆z) and b(t) := v(z+t·∆z), 0 ≤t≤1, and use the Mean
Value Theorem for a(t) and b(t).
5. Let
f(z) = (¯z2/z),if z6= 0,
0,if z= 0.
Show that the real and imaginary parts of fsatisfy the Cauchy-Riemann equations
at z= 0. Is fholomorphic at z= 0?
6. Ahlfors, Ch. 1 Section 2.1 page 15 problem 2: Prove that the points a1,a2,a3are
vertices of an equilateral triangle if and only if
a2
1+a2
2+a2
3=a1a2+a2a3+a3a1.
7. (a) Show that if A6= 0, the set of points
{(x, y)∈R2:A(x2+y2) + Bx +Cy +D= 0
is either empty or a circle. Determine the center and the radius. What
happens when A= 0?
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