MAT 3121 Assignment 1: Due January 27, 2014
Note: You have to justify all computations.
1. Prove the following:
(a) arg(z) arg(z)
(mod 2)
(b) arg(z/w) arg(z) arg(w)
(mod 2)
(c) |z| = 0 if and only if z = 0.
2. Prove Lagranges identity:
2
n
z k wk
k=1
n
n
MAT 3121 Assignment 4: Due April 2, 2014
Note: You have to justify all computations.
1. Show
1
n=0 n2 +z 2
that the series
converges on the set C cfw_z = ni | n Z;
that the convergence is uniform and absolute on each closed disk in this region.
2. By ad
MAT 3121 Assignment 3: Due March 19, 2014
Note: You have to justify all computations.
1. Show
A = A C A;
A = A A;
A is closed if and only if A = A;
If A C and C is closed, then A C.
2. Evaluate the following integrals without performing an explicit co
1
Solutions of Final Exam of Math 3121, 2013, 1st Version
Problem 1. (15 points) Determine if the following maps are homomorphisms
of groups (no reasons needed)
(1). : R R ,
(a) = e3a .
(2). : R R, (a) = ea
(3). : R R, (a) = 2014a
(4). : R R , (a) = 2014a
1
Important Concepts and Theorems Covered in the Final
The nal exam will cover sections 1,4,5,6,8,9,10,13,14,16, 18,19,26. The following notions and theorems are required.
Section 1. n-th root of unity, Un (page 18) ,
Section 4. Group (Denition
4.1). Abel
MAT 3121 Assignment 2: Due February 10, 2014
Note: You have to justify all computations.
1. Let (t) = t(1 + i) and (t) = t + it2 be two paths.
(a) Show that
with end-points xed via a C 2 -homotopy.
(b) Let f (x + iy) = x2 + iy 2 . Compute
f
f.
and
Is it
Midterm Exam MAT 3121
Wednesday October 26th, 2005
Solutions
Problem 1: [12pts] For every complex number z = 1 set
f (z) =
iz 1
(z + 1)2
(1) Solve the equation f (z) = z in C.
Notice that i is a solution, and f (z) = z is equivalent to the
equation z 3 +