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Homework Set 2, Math 208, due Wednesday January 20, 2016 in class.

1. (a) SupposeP

n=0an,

Pm=0bmare absolutely convergent series of real

or complex numbers. Show that

Xn=0

an

Xm=0

bm

=

Xk=0

Xm+n=k

anbm

That is, prove that the right-hand side converges absolutely and showequality of both sides. Can you drop the assumption of absolute con-vergence?

(b) Use (a) to justify the proof ofexey =ex+y based on power series.

2. Let(x, y) = (a(x, y), b(x, y)) beC1(U) whereUis some neighborhoodof (x0, y0) in R

2. Assume (x0, y0) 6= (0, 0).

(a) Let z= x +iy, z0 = x0+iy0 and set

F(x, y) = (z z0)(a+ib)(x, y)

with complex multiplication on the right-hand side (we identify R2

with C as usual). Show that F C1(U) and thatF(U) contains a diskaround (0, 0) in the plane. Hint: Use a theorem from Math 207.

(b) Let N 1 be an integer. Show that there exists C1(V) with

Vsome neighborhood of (x0, y0) so that = N

on V (where N

is inthe sense of powers of a complex number).

(c) Using the notation of (a), (b) define

G(x, y) = (z z0)N(a+ib)(x, y)

Show that G(U) contains a disk around (0, 0).

3. Let p(z) be a non-constant polynomial over C. Show that p is an openmapping. Use this to reprove the Fundamental Theorem of Algebra.

4. Let f(z) =Pn=0an(z z0)

n be a power series with radius of conver-

gence 0 < R . Let A be a complex d d matrix with operatornorm kA z0Ik < Rin C

d (withIthe identity matrix). Show that thematrix-valued series

Xn=0

an(A z0I)n

1

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converges absolutely in the operator norm. We definef(A) to be given

by this series. What does this mean forf(z) = z1

with |z 1|