MATH 122A HW 1 SOLUTIONS
Problem 1. [1.2.1a] Verify that ( 2 ) (1 2) = 2.
( 2 ) (1 2)
2 + 2 2
Problem 2. [1.2.2] Show that
a. (z) = (z);
b. (z) = (z).
a. Let z = x + y. Then (z) = (x y) = y. Also, (z) = y. Th
MATH 122A HW 2 SOLUTIONS
Problem 1. [1.5.1(b)(d)]
b. Notice that
4 + 4 1
Thus z = z.
(2 + )2
Problem 2. [1.5.2]
a. Let S = cfw_z C | ( ) = 2. Let z S. Then ( ) = 2 and thus () = 2. Hence (z
MATH 122A HW 4 SOLUTIONS
Problem 1. [1.14.1] Find a domain in the z plane whose image under the transformation w = z 2 is the square
domain bounded by u = 1, u = 2, v = 1 and v = 2.
Solution. Notice that the preimage of the four given lines are
MATH 122A HW 3 SOLUTIONS
Problem 1. [1.10.1(a)] Find the square roots of 2.
for k Z. Thus
(2) 2 =
Thus the two roots are c0 = 1 + and c1 = (1 + ).
Problem 2. [1.10.2(b)] Find all the roots of (8
MATH 122A HW 5 SOLUTIONS
Problem 1. [2.20.2] Show that
a. a polynomial
P (z) = a0 + a1 z + . . . + an z n an = 0
of degree n (n 1) is dierentiable everywhere, with derivative
P (z) = a1 + 2a2 z + . . . + nan z n1 ;
b. the coecients in the polyn
Name . Perm .
Second Test Math. 122A, summer session A, 2016, Akemann
You may use your calculator, but you must not store notes on it. All action (points, sets, and whatever)
takes place in the complex plane C.
 1a. State the Cauchy-Riemann differentia
Math. 122A, summer 2016, Chuck Akemann, First test
The complex numbers will be denoted by C. U will denote an open region in C.
1. [14 ] Complete the following definitions.
a. If is a positive number, the neighborhood of a point z0 C is
b. A function f :
MATH 122A HW 10 SOLUTIONS
Problem 1. [4.49.1(a)(b)(c)(f)] Apply the Cauchy-Goursat theorem to show that
f (z) dz = 0
when the contour C is the unit circle |z| = 1 in either direction, and when
f (z) =
f (z) = zez ;
f (z) =