Math 814 Final Exam Solutions
1. Let f be a nonconstant entire function. Show that f (C) is dense in C.
First suppose f is a polynomial. If a C, the nonconstant polynomial f (z ) a
has a root, by the Fundamental Theorem of Algebra. Hence if f is a polynom
Math 814 HW 1
September 19, 2007
p.4, Exercise 1. Prove that |z | |w| |z w| and give conditions for equality. It is equivalent to prove that |z | |w|2 |z w|2 , which, after multiplying out and cancelling, amounts to proving that 2|z |w| (z w + z w). (Note
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
Math 814 HW 2
September 29, 2007
p. 43: 1,4,6,13,15, p. 54 1, 3 (cos z only). u(x, y ) = x3 3xy 2 , u(x, y ) = x/(x2 + y 2 ), p.43, Exercise 1. Show that the function f (z ) = |z |2 = x2 + y 2 has a derivative only at the origin. On the region U = C cfw_0
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
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