Here you will find your syllabus for paper C13 - Complex analysis. My notes , assignments and tutorial sheets will be based on this syllabus. Text book for this paper is Complex Variables and Applications by Brown and Churchill (8th Edition, McGraw Publications). For suggested reading Bak and Newman's Complex Analysis (Springer, 2nd Edition) has been included.
C13 Complex Analysis (Under Graduate )
Limits, Limits involving the point at infinity, continuity, properties of complex numbers, regions in the complex plane, functions of complex variable, mappings. Derivatives, differentiation formulas, Cauchy-Riemann equations, sufficient conditions for differentiability.
[1]: Chapter 1 (Section 11), Chapter 2 (Section 12, 13) Chapter 2 (Sections 15, 16, 17,18, 19, 20, 21, 22)
Analytic functions, examples of analytic functions, exponential function, logarithmic function, trigonometric function, derivatives of functions, definite integrals of functions.
[1]: Chapter 2 (Sections 24, 25), Chapter 3 (Sections 29, 30, 34),Chapter 4 (Section 37,38)
Contours, Contour integrals and its examples, upper bounds for moduli of contour integrals.
[1]: Chapter 4 (Section 39, 40, 41, 43)
Antiderivatives, proof of antiderivative theorem, Cauchy-Goursat theorem, Cauchy integral formula. An extension of Cauchy integral formula, consequences of Cauchy integral formula, Liouville’s theorem and the fundamental theorem of algebra.
[1]: Chapter 4 (Sections 44, 45, 46, 50) , Chapter 4 (Sections 51, 52, 53)
Convergence of sequences and series, Taylor series and its examples, Laurent series and its examples, absolute and uniform convergence of power series, uniqueness of series representations of power series.
[1]: Chapter 5 (Sections 55, 56, 57, 58, 59, 60, 62, 63, 66)
Isolated singular points, residues, Cauchy’s residue theorem, residue at infinity. Types of isolated singular points, residues at poles and its examples, definite integrals involving sines and cosines.
[1]: Chapter 6 (Sections 68, 69, 70, 71, 72, 73, 74), Chapter 7 (Section 85).
REFERENCES
James Ward Brown and Ruel V. Churchill, Complex Variables and Applications (Eighth Edition), McGraw – Hill International Edition, 2009.
SUGGESTED READING
Joseph Bak and Donald J. Newman, Complex analysis (2nd Edition), Undergraduate Texts in Mathematics, Springer-Verlag New York, Inc., New York, 1997.
This course assumes that you are familiar with real analysis in one and more variable. It is assumed that you know that the set of complex number is a metric space with respect to euclidean metric. Although two lectures will be devoted to revise your school days complex numbers but an understanding of metric space will make our task of dealing with limit, continuity much easier and we will find that it is similar to real analysis. The dramatic events unfold when we reach to differentiation. We will see that differentiation of complex functions gives some unexpected surprises. Surprisingly many questions whose answers were supposed to be coming from differentiation, get easier if we travel through the beautiful and dense jungle of complex integration integration. Many real integrals are solved by using complex integration.Thus for real world you have to visit imaginary world.
My favourite book is Complex Analysis by Ahlfors, a master of the subject. There are many books on this subject having different flavours. I will suggest you to follow the easiest root of Brown and Churchill.
I am teaching this course from the very first year when it was introduced except with gap of one year when I was on sabbatical (2016-2017). I found that most of the students did not get the finer points of the subject which distinguish it from real analysis. My feeling is that they have not been taught properly ( here , of course, I am talking about sincere students). Another reason is that they missed many subtle points of their courses of real analysis and metric space.
Warning: Some of you may be thinking that since there is only practicals which I will not be taking then you are mistaken. This year your marks in practical will be dependent on your knowledge of theory part. Be careful and alert. Thus theory and practicals are integrated.
I will post my time-table here as soon as I get it. I will be following time-table strictly and religiously, it means if as per my time-table class is on the opening day then it will be held on opening day. Please ensure that you are present in my class from day one.
IMPORTANT: It is compulsory to come in the class with a copy of your prescribed text book:
Complex Variables and Applications by Brown and Churchill, 8th Edition ( Indian Reprint is easily available in the market)
WARNING: Xerox copy is strictly prohibited.
If someone can not afford or purchase the book , please contact me.