Math 220A Complex Analysis Solutions to Homework #1 Prof: Lei Ni TA: Kevin McGown
Conway, Page 4, Problem 2. Basic geometry in R2 says that we have equality in the expression |z1 + + zn | |z1 | + + |zn | if and only if all the zi lie on the same ray with
Math 814 Exam 1 Due: Nov 7, 2007
You may consult books, homework solutions or the instructor, but no one else. All work must be your own.
1. Let f (z ) = z 2 +1. Find equations for and sketch the curves obtained as images of horizontal and vertical lines
Homework 5
Stephen Taylor June 6, 2005
Page 44 16. Find an open connected set G C and two continuous functions f and g dened on G such that f (z )2 = g (z )2 = 1 z 2 for all z G. Can you make G maximal? Are f and g analytic? We dene f (z ) 1 z2 g (z ) 1 z
Homework 4
Stephen Taylor June 4, 2005
Page 43 44 1. Show that f (z ) = |z |2 = x2 + y 2 has a derivative only at the origin. We rst note a theorem from Complex Variables and Applications by Brown: Theorem 1. Let the function f (z ) = u(x, y ) + iv (x, y
Homework II
Stephen Taylor September 27, 2005
Page 10 : 4. Let be a circle lying in S . Then there is a unique plane P in R3 such that P S = . Recall from analytic geometry that P = cfw_(x1 , x2 , x3 ) : x1 1 + x2 2 + x3 3 = l where (1 , 2 , 3 ) is a vect
Math 814 HW 3
October 16, 2007
p. 54: 9, 14, 18, 24, 25, 26 p.54, Exercise 9. If T z = preserve the unit circle.
az +b , cz +d
nd necessary and sufcient conditions for T to
T preserves the unit circle iff |aei + b| = |cei + d|, for all [0, 2 ). Squaring b
Homework I
Stephen Taylor April 28, 2005
Page 3 : z 6. Let R(z ) be a rational function of z . Show that R(z ) = R() if all the coecients in R(z ) are real. Since R(z ) is a rational function of z it is in the form a0 + a1 z + + an z n b0 + b 1 z + + b m
Math 814 HW 5
December 11, 2007
p. 87: 6, 7 p. 96: 8a, 10, 11 p. 110: 1bchi, 5,13. p.87, no. 6. Let f be analytic on D = B (0, 1) and suppose |f (z )| 1 on D. Show that |f (0)| 1. Proof: Let 0 < r < 1 and let r (t) = reit for 0 t 2 . By the Cauchy Integra
Math 814 HW 4
November 7, 2007
p. 74: 5, 6, 7, 9cd, 12, 13, 14. Exercise 5. Give the power series expansion of Log z about z = i and nd its radius of convergence. For any nonzero a C, we have 1 1 1 = = z a 1 + z a a
n=0
(1)n (z a)n , an+1
with radius of c
Exercise Solutions for Complex Analysis
Stephen Taylor April 1, 2005
2
Chapter 1
The Complex Number System
1.1 Pages 2-3
1. Find the real and imaginary parts for each of the following: In the following let the complex variable z = x + iy for cfw_x, y R (
Extra Problems Sheet 1
Stephen Taylor April 28, 2005
1. Consider the complex number z0 =
3+5i 64 3i
(a) Write z0 in the standard form x + iy 3 + 5i 6 + 4 3i 14 3 + 42i 3 i = = + 84 6 2 6 4 3i 6 + 4 3i (b) Find the conjugate of z0 z= (c) Find the modulus
Extra Problems Sheet 2
Stephen Taylor May 11, 2005
1. If |a| < 1 and |b| < 1 show that ab <1 1 ab Proof : Dene a function f (z ) az 1 az
where a and z are complex valued and |a| < 1. Lemma 1. f (z ) is analytic on the open unit disk. We compute f (z ) =
Extra Problems Sheet 3
Stephen Taylor May 4, 2005
1. Dene f : C cfw_ C cfw_ by f (z ) = az + b cz + d
where a, b, c, d are xed complex numbers. Show that f is one-to-one and onto if ad bc = 0. What happens if ad bc = 0? One to one : Suppose f (z ) = f (w)
Extra Problems Sheet 4
Stephen Taylor May 6, 2005
1. Let f (z ) = x2 + y 2 + i2xy (a) Using the denition of dierentiability, determine where f is dierentiable. We note that the denition of dierentiability states Denition 1. If G is an open set in C and f
Extra Problems Sheet 5
Stephen Taylor June 6, 2005
1. The notion of analyticity requires that the function f (x, y ) = u(x, y )+ iv (x, y ) can be written in terms of z = x + iy alone, without using z = x iy . To make this more explicit, we can introduce
Extra Problems Sheet 6
Stephen Taylor May 10, 2005
1. Show that a Mbius transformation T (z ) can have at most two xed points o in the complex plane unless T (z ) z . For a0 , b0 , c0 , d0 C, let T (z ) a0 z + b0 c0 z + d0
be a Mbius transformation. Since
Extra Problems Sheet 7
Stephen Taylor May 15, 2005
1. Prove that if z1 , z2 , z3 are distinct points and w1 , w2 , w3 are distinct points, then the Mbius transformation T satisfying T (z1 ) = w1 , T (z2 ) = w2 , T (z3 ) = o w3 is unique. A Mbius transform
Extra Problems Sheet 9
Stephen Taylor May 20, 2005
1. True or false: z dz =
|z |=1 |z |=1
1 dz z
This is a true statement since both integrals have the value 2i.
2. If is the vertical line segment from z = R > 0 to z = R + 2i, then show that 2e3R e3z dz
Extra Problems Sheet 11
Stephen Taylor May 25, 2005
1. For each of the following, determine whether the statement is true or false. Justify your answers with a brief explanation or counterexample. (a) If f is a nonconstant entire function, then |f (z )| i
Extra Problems Sheet 12
Stephen Taylor June 2, 2005
1. For each of the following functions, classify the behavior at (i.e., analytic, a zero of order n, pole of order m, or an essential singularity) (a) f (z ) =
z 1 z +1 .
We consider f (z 1 ) = 1z 1+z
wh
Extra Problems Sheet 13
Stephen Taylor June 3, 2005
1 (a). Evaluate
0
x4
x2 dx + x2 + 1
We note this function has simple poles at ei/3 and e2i/3 . Since the integrand is even, we nd
0
x4
x2 1 dx = + x2 + 1 2
R
x4
x2 dx + x2 + 1
(1)
We now dene = Reit wh
Complex Analysis Final
Stephen Taylor June 23, 2005
1. For each of the following, determine whether the statement is true or false. Justify your answers with a brief explanation when your response is true and a counterexample when your response is false.
Math 220A Complex Analysis Solutions to Homework #2 Prof: Lei Ni TA: Kevin McGown Conway, Page 14, Problem 11. Parts of what follows are adapted from the text Modular Functions and Dirichlet Series in Number Theory by Tom Apostol. There are shorter proofs
Math 220A Complex Analysis Solutions to Homework #3 Prof: Lei Ni TA: Kevin McGown
Conway, Page 24, Problem 5. Let X be the set of all bounded sequences of complex numbers together with the metric d induced by the sup norm x = supcfw_|xk | : k Z+ , i.e. d
Math 220A Complex Analysis Solutions to Homework #4 Prof: Lei Ni TA: Kevin McGown Conway, Page 33, Problem 7. Show that the radius of convergence of the power series (-1)n n(n+1) z n n=1 is 1, and discuss convergence for z = 1, -1,and i. Proof. The sequen
Math 220A Complex Analysis Solutions to Homework #5 Prof: Lei Ni TA: Kevin McGown
Conway, Page 54, Problem 7. If T (z ) = az + b , cz + d
then nd z2 , z3 , z4 (in terms of a,b,c,d) such that T (z ) = (z, z2 , z3 , z4 ). Proof. We have T 1 (z ) = and we co
Homework 6
Stephen Taylor June 4, 2005
Page 55 :
+b 9. If T (z ) = az+d , nd necessary and sucient conditions that T () = where cz is the unit circle cfw_z : |z | = 1.
We note the general form of a function that maps the unit circle to the unit circle is
Homework 7
Stephen Taylor June 4, 2005
Pages 54 57 : 6. Evaluate the following cross ratios. We rst note that the cross ratio is dened to be Denition 1. If z1 C then (z1 , z2 , z3 , z4 ) is the image of z1 under the Mbius o transformation which takes z2 1
Homework 8
Stephen Taylor June 4, 2005
Pages 67 68 : 5. Let (t) = exp(i 1)t1 ) for 0 < t 1 and (0) = 0. Show that is a rectiable path and nd V (y ). Give a rough sketch of the trace of . We apply Proposition (1.3) noting that is a piecewise smooth functio
Homework 9
Stephen Taylor May 20, 2005
Pages 74 75 : 5. Give the power series expansion of log z about z = i and nd its radius of convergence. Letting f (z ) = log z we nd that its n-th derivative is given by f (n) (z ) = (1)n1 (n 1)!z n Using the standar