NOTES ON HISTORY OF GEOMETRY
We consider only the plane under different geometries. We use PG(F) for the
projective plane over a field F, and AG(F) for the affine plane over F.
Projective plane: In ho
MATH 4X3/6X3 ASSIGNMENT #2 SOLUTIONS
FEBRUARY 9, 2009
Question #1: Let log z be the inverse of the exponential function ez in the 1 simply connected set = C n R that satises log 1 = 0. Then log0 z = z
Notes on the History of Number Theory
0. Preliminaries: Field
Before we can consider classical number theory, and its various generalizations to what
are called algebraic number fields, we need to con
MATH 4X3/6X3 ASSIGNMENT #1 SOLUTIONS
JANUARY 26, 2009
Question #1: We have f (z) = z jzj 0=
2
1
jzj
2
= z2z
z3z2;
and so the Cauchy-Riemann equations hold for those z satisfying @ @ @ 2 z2z z 3 z 2 =
MATH 4X3/6X3 ASSIGNMENT #3 SOLUTIONS
MARCH 9, 2009
(z) Question #1: Let g (z) = fez . Then g : D ! D is holomorphic in D and g (0) = 0. By the open mapping theorem (or the maximum modulus principle) w
MATH 4X3/6X3 ASSIGNMENT #4 SOLUTIONS
MARCH 30, 2009
Question #1: We have (0.1) Since
0
:T!
and z2 :
jf (z)
g (z)j < jf (z)j ;
0 is compact in and is convex, there is a convex set 0 with 0 . Then any f
MATH 4X3/6X3 ASSIGNMENT #2 SOLUTIONS
FEBRUARY 10, 2011
Question #1: Suppose that f : C ! C is holomorphic and has polynomial
N
growth at innity, i.e. jf (z)j
C 1 + jzj
for some positive constant C and
MATH 4X3/6X3 ASSIGNMENT #1 SOLUTIONS
JANUARY 20, 2011
Question #1: We have
2
f (z) = z jzj
1
2
= z2z
jzj
z3z2;
and so the Cauchy-Riemann equations hold for those z satisfying
0=
@
@
f (z) =
z2z
@z
@z
MATH 4X3/6X3 ASSIGNMENT #3 SOLUTIONS
MARCH 17, 2011
Question #1: Let f and g be one-to-one onto holomorphic maps from an open
set
to the unit disk D. Let a be a point in
and let c = f (a) and d = g (a
MATH 4X3/6X3 ASSIGNMENT #4 SOLUTIONS
MARCH 31, 2011
Question #1: For 0 < " < 1 let
"
(s) =
Z
1
"
e t ts
1
dt;
s 2 C:
"
Let T be a triangle path in C. Since the integrand g (s; t)
e t ts 1 is jointly
c
/ Name
[29 @_
< Question 2 about Gaussian integers:
(1) Prove that N (AB) = N (A)N (B)
(2) Using (1), show that N (A) = 1 if and only if A is a unit.
(3) Determine which one of 3 + i, 4 + i, and 5 + i