Math 220 Midterm
Jim Agler
November 14, 2011
1. Let and be the two polygonal paths [1, i] and [1, 1+ i, i] and let f (z ) =|z|2 .
Compute and . What do these calculations imply about f ?
2. Prove that there is no analytic function on cfw_z C | |z| < 1 sat
Math 220A Midterm Solutions
Jim Agler
1. Compute
zez dz where (t) = eit , 0 t .
Solution. F (z ) = (z 1)ez is a primitive of zez . Therefore, as (0) = 1 and ( ) = 1,
zez dz = F (1) F (1)
= (1 1)e1 (1 1)e1
= 2e1 .
2. Prove the Fundamental Theorem of Algebr
Math 220A Homework 2 Solutions
Jim Agler
10. Let G be an open set in C.
(a)Show that the product rule for z and z holds for products of C 1
functions on G.
2
(b) Show that if f is analytic on G, then z z |f (z )|2 = |f (z )|2 for all
z G.
(c) Show that if
Math 220A Homework 3 Solutions
Jim Agler
22. (This exercise makes it clear where z and z come from). Let f (z ) be a complex
valued function dened on a neighborhood D of z0 C. Let f (z ) = u(z ) + iv (z ) where u
and v are real valued on D. By setting z =
Math 220A Homework 4 Solutions
Jim Agler
26. (#1 pg. 73 Conway). Prove the assertion made in Proposition 2.1 (pg.
68) that g is continuous.
Solution. We wish to show that if g : [a, b] [c, d] C is a continuous
function and g : [c, d] C is dened by
b
g (t)
Math 220A Homework 1 Solutions
Jim Agler
October 11, 2013
1. Show that if f is a dierentiable complex valued function dened on a
neighborhood of z0 C and b and c are complex numbers such that
|f (z ) (b + c(z z0 )| = o(|z z0 )|),
(0.1)
then b = f (z0 ) an
Math 220A Homework 5 Solutions
Jim Agler
38. (Exercise # 6 pg. 87 Conway). Let f be analytic on D = B (0, 1) and suppose |f (z )| 1
for |z | < 1. Show |f (0)| 1.
Solution 1. Fix r < 1 and let = reit , 0 t 2 . By the Cauchy Integral Formula
(cf. Corollary