FACULTY OF SCIENCE
SCHOOL OF MATHEMATICS AND
STATISTICS
MATH2621
HIGHER COMPLEX ANALYSIS
Session 2, 2014
:
Cricos provider number 00098G
MATH2621 Course Outline
Information about the course
Course Authority: Professor Michael Cowling
Lecturer: Professor M
LECTURE 31
Fractional linear transformations
In this lecture, we study a particular type of conformal mapping: fractional linear
transformations. This are easier than general conformal mappings because we can use
ideas from linear algebra to simplify comp
LECTURE 20
Laurent series
In this lecture, we prove Laurents theorem about representing holomorphic functions in
an annulus by a series, now called a Laurent series, and we give some examples of Laurent
series.
An annulus is a set of the form cfw_z C : R1
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 35
Conformal mappings, harmonic functions, and aerofoils
In this lecture, we examine harmonic functions and the Dirichlet problem
1. The Dirichlet problem
Let be a domain in R2 , let B be a subset of its boundary , and let b : B R be
continuous. T
LECTURE 28
The Laplace Transform
In this lecture, we introduce the Laplace transformation. The first step is to define the
class of functions on which it will act.
1. Functions of exponential type
Definition 28.1 (Functions of exponential type). Suppose t
LECTURE 29
Applications to differential and integral equations
We may use the Laplace transform to solve differential and integral equations on [0, ).
1. Differential equations
The Laplace transform may be used to solve differential equations on [0, +).
E
LECTURE 13
Cauchys integral formula
Cauchys integral formula is perhaps the most important formulae in the theory of
functions of a complex variable.
In this lecture, we establish Cauchys integral formula, and prove one of its corollaries.
We also apply i
LECTURE 22
Residues and the residue theorem
In this lecture, we define residues, and prove Cauchys residue theorem. We show how
to find residues. This will enable us to find integrals over closed contours efficiently.
1. Residues
Definition 22.1. Suppose
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
LECTURE 21
Singularities
In this lecture, we study holomorphic functions in a punctured ball, and illustrate
this study with some examples. Recall that the punctured ball B (z0 , R) is defined to be
cfw_z C : 0 < |z z0 | < R; we allow R to be .
1. Laurent
LECTURE 14
Contour integrals
In this lecture, we define contour integrals, and compute some examples.
1. Curves and contours
We define a curve in C in much the same way as in R2 . Then a curve t 7 (t), (t) is
replaced by a curve t 7 (t) + i(t). The ideas
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 multivariable calculus.
1. Curves
Definition 13.1. A curve in R2 is a
LECTURE 18
Contour integrals and Moreras theorem
In this lecture, we prove Moreras theorem, which completes a logical circle relating differentiability and independence of contour for integrals. We also compute more
examples of contour integrals.
1. Morer
LECTURE 23
Computing integrals. 1
In this lecture, we give three examples of the use of Cauchys residue theorem to
calculate integrals.
1. Trigonometric integrals
Z
cos()
d.
Exercise 23.1. Evaluate
5 4 cos()
Answer. The first step is to convert this to
LECTURE 12
Inverses of the Exponential, Hyperbolic and Trigonometric
functions
In this lecture, we discuss the inverse functions for the exponential, hyperbolic and
trigonometric functions introduced in the last lecture.
1. Inverse functions
The following
LECTURE 10
Power series
A (complex) power series is an expression of the form
X
an (z z0 )n ,
(1)
n=0
where the centre z0 and the coefficients an are all fixed complex numbers, and the variable
z is complex. We take (z z0 )0 to be 1 for all z, even when z
LECTURE 15
The CauchyGoursat theorem and antiderivatives
In this lecture, we recall the CauchyGoursat theorem, establish some of its corollaries,
and sketch its proof.
1. The CauchyGoursay theorem
We recall one form of the theorem.
Theorem 15.1. Suppose t
LECTURE 26
Interlude: Exchanging integrals and limits
In this lecture, we consider integrals in which the integrand contains a parameter, and
consider the question of whether
Z
Z
lim
F (s, t) ds =
lim F (s, t) ds.
(26.1)
tt0
tt0
In our theorem, we supp
LECTURE 19
Taylor series
We recall some facts about power series and Cauchys integral formula. Then we
review examples of Taylor series.
1. Power series, Taylor series, and Maclaurin series
Definition 19.1. A power series (with centre z0 ) is an expressio
LECTURE 8
Differentiable Functions
In this lecture, we investigate some properties of functions that are differentiable in the
complex sense. Such functions are very special, and have some surprising properties, which
we prove using the CauchyRiemann equa
LECTURE 30
Miscellanea
In this lecture, we prove Liouvilles theorem, which tells us that there are fewer
complex differentiable functions than real differentiable functions, and we use this to prove
the fundamental theorem of algebra. We count the number
LECTURE 34
Miscellanea. 2
In this lecture, we count the number of times a closed curve winds around a point,
and count the number of zeroes of a function inside a closed simple curve. We see that
this helps us to find maxima of |f (z)|, where f is holomor
LECTURE 31
Conformal mappings and harmonic functions
In this lecture, we examine conformal mappings: mappings that preserve form, at
least locally. We then consider the effect of composing harmonic functions and holomorphic
functions.
1. Affine mappings a
LECTURE 11
Exponential, Hyperbolic and Trigonometric functions
1. The exponential function
Definition 11.1. We define the exponential series by the formula
exp(z) =
X
zn
n=0
n!
z C.
It follows from the definition that this is the only power series extensi
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
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