MATH 407-10a, Assignment 1 Guidelines Answer the questions in the space provided; you may write on both sides of the paper. Put the names of all group members in the top right corner. You may append additional sheets as needed. Please staple all sheets to
Sivakumar Example Sheet 4c
M407
1. Let log(z ) denote a branch of the logarithm dened on the region D := C \ cfw_x(1 + i) : x 0. Assume that log(i) = 5i/2 . (i) Evaluate log(1), log(2i), log(1 i), and log(3). (ii) Find the image of the lower-half plane H
Sivakumar Example Sheet 5 In what follows we shall use the following notation: H := cfw_z = x + iy C : x 0, y R
M407
1. Suppose 0 < < 2 is a xed number, and let f (z ) denote the principal branch of the (multivalued) function z z . Find (and sketch) the i
Sivakumar Example Sheet 6
M407
1. Let a < b be real numbers, and suppose that G, H : [a, b] C are continuous on [a, b]. Let be a xed complex number. Verify the following statements:
b b b
(i)
a b
(G(t) + H (t) dt =
a b
G(t) dt +
a
H (t) dt.
(ii)
a
(G)(t)
Math 407 1. (15) Define the following:
Solutions to Exam 1
February 11, 2003
(a) A boundary point of a set S. The point z0 is a boundary point of a set S if every epsilon neighborhood of z0 contains a point of S and a point not in S. (b) The imaginary par
Math 407
Solutions Exam 2
March 28, 2011
1. (30) Let denote a curve in C , and let (t) = (x(t), y(t) for a t b, be a parametric representation for . Suppose that each of the coordinate functions of is smooth. That is, both x and y have continuous derivati
Math 407 1. (20) Define the following: (a)
Solutions Exam 2
March 6, 2003
f (z) dz, where C is any contour. Suppose the contour is parametrized by the function z (t) = x (t) + iv (t) for a t b, and f (z) = u + iv. Then
C b C
f (z) dz =
a
u (x (t) , y (t)
Math 407
Exam 3 Solutions
April 27, 2011
1. (25) Show that the series
n=0
zn converges normally to ez in the complex plane. n!
If F is any compact subset of C , then there is an R > 0 such that F cfw_z : |z| R. We'll show that the series converges uniform
Math 407 1. (12) Define the following: (a)
Solutions to Exam 3
April 24, 2003
f (z) dz, where is a smooth curve with finite length. Let f = u + iv, and let be parmetrized by = (x (t) , y (t) = x (t) + iy (t) for a t b. Then
b
f (z) dz =
a
dx dy u -v dt d
Sivakumar Example Sheet 4b
M407
1. Suppose that n is a xed positive integer. Show that cos(n) can be expressed as a polynomial
n
in cos ; more precisely, there exist real numbers a0 , . . . , an such that cos(n) =
k=0
ak (cos )k .
(Use strong induction: T
Sivakumar Example Sheet 4a
M407
1. Let m be a xed positive integer. Suppose that a0 , . . . , am are complex numbers and am = 0.
m
Dene P (z ) :=
k=0
ak z k , z C. Show that lim P (z ) = .
z
2. Dene T (z ) :=
zi , z+i
z = i .
(i) Show that T is a one-to-
Sivakumar Example Sheet 3b
M407
1. Suppose that z is a complex number and that is a xed positive integer. Show that the series
n z n is absolutely convergent for |z | < 1 and divergent otherwise.
n=0
2. Show that the series
zn is absolutely convergent for
MATH 407-10a, Assignment 2 Guidelines Answer the questions in the space provided; you may write on both sides of the paper. Put the names of all group members in the top right corner. You may append additional sheets as needed. Please staple all sheets to
MATH 407-10a, Assignment 3 Guidelines Answer the questions in the space provided; you may write on both sides of the paper. Put the names of all group members in the top right corner. You may append additional sheets as needed. Please staple all sheets to
MATH 407-10a, Assignment 4 Guidelines Answer the questions in the space provided; you may write on both sides of the paper. Put the names of all group members in the top right corner. You may append additional sheets as needed. Please staple all sheets to
MATH 407-10a, Assignment 5 Guidelines Answer the questions in the space provided; you may write on both sides of the paper. Put the names of all group members in the top right corner. You may append additional sheets as needed. Please staple all sheets to
MATH 407-10a, Assignment 6 Guidelines Answer the questions in the space provided; you may write on both sides of the paper. Put the names of all group members in the top right corner. You may append additional sheets as needed. Please staple all sheets to
MATH 407-10a, Assignment 7 Guidelines Answer the questions in the space provided; you may write on both sides of the paper. Put the names of all group members in the top right corner. You may append additional sheets as needed. Please staple all sheets to
Sivakumar Example Sheet 1 1. Find a polar representation (i.e., modulusamplitude form) of the complex number z = What is Arg(z ) ? 2. Show that |z | = 1 if and only if 1/z = z . 3. Suppose z and w are nonzero complex numbers. Show that zw is nonzero. 4. S
Sivakumar Example Sheet 2
M407
1. (a) Suppose that R is a xed positive number. Show that the solutions of the equation z 2 = R are given by R. (This shows that there is no loss of consistency when passing to complex roots of positive real numbers.) (b) So
Sivakumar Example Sheet 3a 1. Suppose that is a xed real number. Evaluate the following limits: cos(n) + i sin(n) n (ii) lim n 1 cos i sin n n (i) lim
n
M407
n
2. Show that the limit of a convergent sequence of complex numbers is unique. 3. Suppose that
Math 407
Final Exam Solutions Every question is worth 30 points.
May 10, 2011
1. Use the definition of cos z to answer the following questions. a. Find all zeros of cos z, and their orders. cos z = eiz + e-iz = 0 = eiz = -e-iz = e2iz = -1 = ei+2ki 2 2iz =