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 homogeneous coordinates, the transformations are
x1 = a11
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 and by the coincidence principle, (0.1) log (1 + 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 consider the concepts of group and field.
The general conc
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 = z 2 z 3 2z = z 2 1 2 jzj ; f (z) = @z @z @z 1 i.e. for
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) we have g : D ! D unless g is constant in which case it
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 2 H ( ) has only nitely many zeroes in . From (0.1) we
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
positive integer N . We show that f (z) is a polynomia
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
@
z3z2 = z2
@z
1
p
2
i.e. for z at the origin or in the
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).
Find a relation between f and g that involves a; c;
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
continuous in (s; t), it follows that g is uniformly con
/ 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 is a Gaussian prime.
For the other two, factor them in