THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
MATH2621
Higher Complex Analysis
Problem Sheet
S2, 2017
Good solutions to these problems include reasons. A few problems may require
extensions of ideas or definitions in the course; t
THE UNIVERSITY OF NEW SOUTH WALES
School Of Mathematics and Statistics
MATH2621
Higher Complex Analysis
Class Test 1 Sample
S2, 2016
You have 25 minutes for this test. Write your answers, together with your name
and student number, on blank sheets of pape
THE UNIVERSITY OF NEW SOUTH WALES
School Of Mathematics and Statistics
MATH2621
Higher Complex Analysis
Class Test 3 Sample
S2, 2016
You have 25 minutes for this test. Write your answers, together with your name
and student number, on blank sheets of pape
THE UNIVERSITY OF NEW SOUTH WALES
School Of Mathematics and Statistics
MATH2621
Higher Complex Analysis
Class Test 2 Sample
S2, 2016
You have 25 minutes for this test. Write your answers, together with your name
and student number, on blank sheets of pape
THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
November 2015
MATH2621
Higher Complex Analysis
(1) TIME ALLOWED 2 hours
(2) TOTAL NUMBER OF QUESTIONS 4
(3) ANSWER ALL QUESTIONS
(4) THE QUESTIONS ARE OF EQUAL VALUE
(5) ANSWER EACH QU
Technical Proofs
Math22221
August 18, 2015
1
Local existence and uniqueness
We consider an initial-value problem for a nonlinear system of ODEs,
dx
= F (x, t) for all t, with x(0) = x0 .
dt
(1)
Since x(t) is a solution iff it satisfies the nonlinear Volte
THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
November 2014
MATH2621
Higher Complex Analysis
(1) TIME ALLOWED 2 hours
(2) TOTAL NUMBER OF QUESTIONS 4
(3) ANSWER ALL QUESTIONS
(4) THE QUESTIONS ARE OF EQUAL VALUE
(5) ANSWER EACH QU
THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
NOVEMBER, 2012
MATH2620
HIGHER COMPLEX ANALYSIS
(1) TIME ALLOWED 2 hours
(2) TOTAL NUMBER OF QUESTIONS 4
(3) ANSWER ALL QUESTIONS
(4) THE QUESTIONS ARE OF EQUAL VALUE
(5) ANSWER EACH Q
THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
NOVEMBER, 2011
MATH2620
HIGHER COMPLEX ANALYSIS
(1) TIME ALLOWED 2 hours
(2) TOTAL NUMBER OF QUESTIONS 4
(3) ANSWER ALL QUESTIONS
(4) THE QUESTIONS ARE OF EQUAL VALUE
(5) ANSWER EACH Q
Higher Complex Analysis. XXVI
Computing sums
September 20, 2016
1 / 26
Today?
In this lecture, we show how residue calculus can be used to
evaluate sums. We will also see the Riemann zeta function, a focus
of current mathematical research.
2 / 26
A square
Higher Complex Analysis. XXXIII
Fractional linear transformations
October 13, 2016
1 / 24
Today?
In this lecture, we study a particular type of conformal mapping:
fractional linear transformations. These are easier to handle than
general conformal mapping
Higher Complex Analysis. XXVIII
The Fourier transformation
October 4, 2016
1 / 23
Today?
In this lecture, we introduce the Fourier transform fb of an
integrable function f on R; we then compute some examples.
2 / 23
Today?
In this lecture, we introduce th
THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
MATH2621
Higher Complex Analysis
Problem Sheet
S2, 2015
Revision exercises.
1. Show that |ei 1| = 2 |sin(/2)| for all R
(a) using the geometry of the triangle with vertices 0, 1, and t
LECTURE 13
Paths and path integrals
In this lecture, we define paths and path integrals, and see a key theorem about
these. This material should be familiar to students who have studied multi-variable
calculus.
The main question underlying this lecture is
LECTURE 0
Assumed Knowledge
This is a review of basic facts about complex numbers that ought to be familiar:
the definition of complex numbers,
their arithmetic,
Cartesian and polar representations,
the Argand diagram,
de Moivres theorem, and
extracting n
LECTURE 0
Assumed Knowledge
This is a review of basic facts about complex numbers that ought to be familiar:
the definition of complex numbers,
their arithmetic,
Cartesian and polar representations,
the Argand diagram,
de Moivres theorem, and
extracting n
MATH2621 Higher Complex Analysis. IV
Limits and continuity
August 2, 2016
1 / 29
This lecture?
In this lecture, we outline the key ideas and facts about limits and
continuity, as a preliminary to defining differentiability.
2 / 29
This lecture?
In this le
THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
MATH2621
Higher Complex Analysis
Problem Sheet
S2, 2016
Revision exercises.
1. Show that |ei 1| = 2 |sin(/2)| for all R
(a) using the geometry of the triangle with vertices 0, 1, and t
THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
MATH2621
Higher Complex Analysis
Problem Sheet
S2, 2016
Revision exercises.
1. Show that |ei 1| = 2 |sin(/2)| for all R
(a) using the geometry of the triangle with vertices 0, 1, and t
THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
MATH2621
Higher Complex Analysis
Problem Sheet
S2, 2016
Revision exercises.
1. Show that |ei 1| = 2 |sin(/2)| for all R
(a) using the geometry of the triangle with vertices 0, 1, and t
THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS
November 2009
MATH2620
Higher Complex Analysis
(1) TIME ALLOWED 2 HOURS
(2) TOTAL NUMBER OF QUESTIONS 4
(3) ANSWER ALL QUESTIONS
(4) THE QUESTIONS ARE OF EQUAL VALUE
(5) ANSWER EACH QUESTION IN A SEP
LECTURE 25
The Theory of Functions
In this lecture, we count the number of times a closed curve winds around a
point, and use this to count the number of zeros and poles of a function. This leads
to some surprising facts about holomorphic functions.
We st
THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
NOVEMBER, 2010
MATH2620
HIGHER COMPLEX ANALYSIS
(1) TIME ALLOWED 2 hours
(2) TOTAL NUMBER OF QUESTIONS 4
(3) ANSWER ALL QUESTIONS
(4) THE QUESTIONS ARE OF EQUAL VALUE
(5) ANSWER EACH Q
THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
MATH 2620
HIGHER COMPLEX ANALYSIS.
Solutions to 2010 Examination.
Question 1.
i) Solving the implied quadratic, we have (z 2)3 = 8, (z 2)3 = 1
whence z = 4, 2 + 2e2i/3 , 2 + 2e2i/3 , 2
THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
MATH 2620
HIGHER COMPLEX ANALYSIS.
Solutions to 2011 Examination.
Question 1.
i)a. DIAGRAM
b. The function represents a dilation (by
and a translation by i.
2), an anticlockwise rotati
Higher Complex Analysis. XXXI
The Dirichlet problem
October 11, 2016
1 / 25
Today?
In this lecture, we discuss the Dirichlet problem. We show that the
solution to this problem in a domain is unique, if it exists, using
the so-called maximum principle; the
Higher Complex Analysis. XXIV
Computing Integrals. II
September 15, 2016
1 / 22
Today?
In this lecture, we compute more definite integrals, and introduce
Jordans Lemma, which can be useful in this context.
2 / 22
Exercise 1
Evaluate
Z
e i x
dx, where R+ .
MATH2621 Higher Complex Analysis. XXIX
The Laplace transformation
October 4, 2016
1 / 26
Today?
In this lecture, we introduce the Laplace transformation L. The
first step is to define the class of functions on which L will act.
Then we define L and give s
MATH2621 Higher Complex Analysis. XVI
Cauchys integral formula
August 30, 2016
1 / 21
This lecture?
In this lecture, we
sketch a proof of the CauchyGoursat Theorem,
state and prove Cauchys integral formula, and
see some applications.
2 / 21
Proof of the C
MATH2621 Higher Complex Analysis. I
Inequalities and sets of complex numbers
July 26, 2016
1 / 24
Information
Lecturer: Professor M. G. Cowling, RC-5113,
[email protected]
2 / 24
Information
Lecturer: Professor M. G. Cowling, RC-5113,
[email protected]