Math 506 Homework 1 Solution
(2) Let n 3 be an integer. Let f (x) and g(x) be polynomials with
real coecients such that the points (f (1), g(1), (f (2), g(2), .,
(f (n), g(n) in R2 are the vertices of a regular n-gon in counterclockwise order. Prove that
ALGEBRAIC CLOSURE OF O[z]
We work over the ring O[z] of germs of analytic functions in z, although
the same statement holds for the ring C[z] of formal power series in z.
Theorem 0.1. Let O([z]) be the quotient eld of O[z]. Then any nite
extension of O([z
ON A THEOREM OF G. POLYA
Theorem 0.1. Suppose that cfw_an : n = 0, 1, . be a sequence of complex
numbers such that
(0.1)
n=0
an+1
an
2
converges. Then the power series
(0.2)
an z n
f (z) =
n=0
converges everywhere and
(0.3)
n=1
1
|zn |2
converges, where z
BLOCHS THEOREM
Lemma 0.1. Let f be analytic in = cfw_|z| < 1 with f (0) = 0 and f (0) =
1. If |f (z)| M for all z , then f () contains the disk |w| ( M + 1
2
M) .
Proof. By Schwartzs lemma, we implicitly have M 1. Let f (z) = z +
n
n=2 an z . Using CIF,
Math 506 Homework Solution
Let f (x, y) O2 [x, y]. Suppose that
(0.1)
ajk xj y k
f (x, y) = (x c1 y)(x c2 y).(x cm y) +
j+k>m
where c1 , c2 , ., cm , ajk C are constants. We call m the multiplicity
of f (x, y) at the origin. Show that if c1 , c2 , ., cm
MODULI OF ANNULI
Let R,r be the annulus r < |z| < R, where R > r 0. Clearly, R,r
and R,r are conformally equivalent for all > 0. On the other hand, we
claim that
(0.1)
1,exp() 1,exp()
=
for all = 1 R+ . This will prove that R,r R ,r if and only if
=
r
r
=
Question 0.1. Let O([z]) be the quotient eld of O[z]. Then any nite
extension of O([z]) is O([z]) itself and given by : O([z]) O([z]) with
(z) = z n for some n > 0.
First let us prove a lemma.
Lemma 0.2. Let f (z, w) = wd + a2 (z)wd2 + . + ad (z) O[z], w]
MORE ON MORERAS THEOREM
Moreras Theorem says that a function f (z) is analytic in an open set
D C if and only if f (z) is continuous in D and
(0.1)
f dz = 0
ABC
for all ABC D.
There is nothing special about triangles. They can be replaced by other
type of
Math 506: Complex Variables
James D. Lewis
Winter Term
Detailed Syllabus
Part I: Single Variables: Review and extensions. CR equations and analyticity; Cauchy-Goursat theorem and Cauchy integral formula, Louivilles theorem;
Moreras theorem; maximum modulu