PMATH 352 Complex Analysis, Assignment 6
Not to hand in
(b) Let f (z ) = 3z 3 2z 2 + z + 1. Let g be the inverse of the restriction of f to a disc
centered at 0. Find the Taylor polynomial of degree 4 centred at 1 for g .
. Find the Laurent se
PMATH 352 Complex Analysis, Assignment 5
Due Fri July 20
1: (a) Let (t) = (2 sin t)eit for 0 t 2 . Let (t) = (t)3 . Sketch (t) and nd its length.
(b) Let (t) = (2 4 sin t)eit for 0 t 2 . Sketch (t) and use the sketch to nd the
winding numbers (, ai ) for
PMATH 352 Complex Analysis, Assignment 4
Not to hand in
1: Let S2 be the unit sphere x2 + y 2 + z 2 = 1. Let : S2 \ cfw_n C be the stereographic
projection from the north pole n = (0, 0, 1) and let : S2 \ cfw_s C be the stereographic
projection from the s
PMATH 352 Complex Analysis, Assignment 3
Due Fri June 15
1: (a) Let f (z ) = ez and let U = x + iy x 0 , 0 y . Sketch f (U ) and state (without
proof) whether it is open, closed, convex, connected, bounded and/or compact.
(b) Let f (z ) = z 3 and let V =
PMATH 352 Complex Analysis, Assignment 2
Due Fri June 1
1: Let f (z ) = z 2 .
(a) Find the images of the lines Re(z ) = 0, 1, 2 and Im(z ) = 0, 1, 2 under the map w = f (z ).
(b) Find the inverse-images of the lines Re(w) = 0, 1, 4 and Im(w) = 0, 2, 8 und
PMATH 352 Complex Analysis, Assignment 1
Due Fri May 18
1: Solve each of the following equations for z C. Express your answers in cartesian form.
(a) i z 2 + (2 + i) z + (7 + i) = 0
(b) z 5 + 4 z = 0
(c) z 8 = (4 + 3i)4
2: For each of the following polyno
Distance Preserving Maps
Theorem: Let U Rk be a non-empty open set. Let f : U Rl . Then f preserves distance if and only if
it is of the form f (x) = Ax + b for some A Mlk (R) with AT A = I and some b Rl .
Proof: First, suppose that f (x) = Ax + b where A
Chapter 9. Laurent Series and Residues
9.1 Note: We have studied power series. We are also interested in series of the form
cn (z a) =
cn (z a) +
cn (z a) c =
cn (z a)n .
cn w +
where we have written w = 1/(z a). If the rst ser
Chapter 8. Power Series
8.1 Denition: A sequence of complex numbers is a function f : cfw_k, k + 1, k + 2 . . . C
where k Z. We usually write f (n) as an and we denote the sequence f by cfw_an nk or
simply by cfw_an . For a C, we say that the sequence cfw
PMATH 352 (1115 - Spring 2011)
Professor: W. Kuo
University of Waterloo
Author: mlbaker <lambertw.com>, with contributions from Jimmy Zhu
Revised: August 2011
2.1 Holomorphic functions . . .
gamma mU< 2: no mono: 0.562 ownmunch
Son 53 H HumansEco oh .838 05 no 3:32 x0085 3650 o: E Bush
.5 3:0; quo wccsu 63830 E. .oo 9 8E : 23m: 03 9: .5309 P: 5on
m=2 m 039,5 moi t :5 .msz BE SEEN :5 H .935? $223 :33 me
Chapter 7. Cauchys Integral Formulas
7.1 Remark: Let U C be open, and let be a path which runs counterclockwise
around the boundary of a closed convex set E U . Recall that Greens theorem (for real
path integrals) states that if u, v : U R are C 1 maps, t
Chapter 6. Integration
6.1 Denition: Let I be an interval in R (I could be open, closed or half-open). Say
I = a, b where a, b R cfw_ with a < b and where and denote either open or
closed brackets depending on whether a and b are open or closed endpoints
Chapter 5. Conformal Maps
5.1 Note: Later on we shall see that every holomorphic function is C , which means that
all partial derivatives of all orders exist (and are continuous). For this chapter we shall
assume that all functions are C 2 , which means t
PMATH 352, FALL 2009
Due: November 27
for z C \ cfw_0, 1, 2. Compute the
z (z 1)(z 2)
Laurent series for f on each of the annuli A(0; 0, 1), A(0; 1, 2) and
A(1; 0, 1).
1. (a) Let f (z ) =
[Dont be afraid of Cauchy products (A2, Q2), if you
Pure Math 352, Assignment 4: Solution Sketch
NOTE: PICTURES ATTACHED IN SEPARATE FILE.
1. (a) We need only to exhibit the homotopies. Reexivity: H (s, t) = 0 (t) shows 0 0 .
Symmetry: if H shows 0 1 , then H1 (s, t) = H (1 s, t) shows i 0 . Transitivity
PMATH 352, FALL 2009
Due: November 16
1. (a) Let V C and (V ) be the family of all closed curves in V . Show
that the relation of homotopy on (V ) is an equivalence relation, i.e.
show for 0 , 1 , 2 in (V ) that
(reexivity) 0 0 ,
Pure Math 352, Assignment 3: Solution Sketch
1. (a) For z = 0 we have sin z =
n=0 (2n+1)! z , and this series converges uniformly
on compact subsets of C \ cfw_0, in particular uniformly on D(0, r). Thus we can
interchange integral and limit a