NPTEL Syllabus
Complex Analysis - Video course COURSE OUTLINE Complex Numbers and Algebra, Spherical representation of extended complex plane, Analytic functions, Harmonic functions, Elementary Elementary functions, Branches of multiple-valued functions, Mappings of elementary functions, Bilinear Bil inear transformations, transformations, Conformal mappings and Computational aspects. Complex i ntegration, Cauchy's integral theorem, Winding number, number, Cauchy's integral formula, Theorems of Morera and Liouvi lle, Maximum-Modulus Maximum-Modulus theorem, P ow er series, Taylor Taylor’s ’s theorem and analytic continuation, Zeros of analytic functions, Hurwitz theorem, Singularities, Laurent’s theorem, CasoratiWei erstrass erstra ss theorem, Argument principle, Theorems of Rouche and GaussLucas, Residue theorem and its applications to evaluate real integrals and to compute the i nverse Laplace Lapl ace transform, transform, Contour integration of functions w ith branch points.
NPTEL http://nptel.iitm.ac.in
Mathematics Additional Reading: J. E. Marsden and M. J. Hoffman, Basic Complex Analysis, Analysis, 3rd edition, W. H. Freeman, 1999.
COURSE DETAIL Lectures
Topic/s
1
Comple Complex x num numbe bers rs,, Algeb Algebra ra in the the comple complex x plan plane, e, Conjug Conjugat ation ion,, modulus and inequalities.
L. V. Ahlfors, Complex Analysis, 3rd edition, McGraw McGraw Hil l, 1979.
Pola Polarr for form m, Pow Power ers s and and root roots s of of com compl plex ex num numbers bers..
John B. Conway, Functions of One Complex Variable, 2nd edition, Narosa, 1980.
2
3
Geome eometr try y in the the com comple plex x plan plane, e, The exte extend nded ed comp complex lex plane plane and the Riemann sphere.
4
Topolo opology gy in C: Int Inter erior ior point points, s, Limit Limit poin points ts,, Ope Open n sets sets,, Clos Closed ed sets, Connected sets, Compact sets.
5
Sequ Sequen ence ces s of of com complex plex num numbe berrs and and conv conver erge genc nce. e.
6
Comple Complex x funct function ions, s, Visua Visualizin lizing g comple complex x funct function ions. s. Limit Limits s of of functions, Continuity.
7
(Com (Comple plex) x) Diffe Differe rent ntiat iation ion and and the the Cauchy Cauchy-R -Riem ieman ann n equ equat ation ions. s.
8
Anal ytic fu functi ons.
9
Harm Harmon onic ic func functtions ions,, Fin Findi ding ng Har Harmonic onic con conju juga gattes. es.
10
Eleme Element ntar ary y Ana Analyt lytic ic fun funct ction ions: s: Expon Exponen enti tial al fun funct ction ion,, Trigonometric functions and their mapping properties.
R. A. Silvermann, Introductory Complex Analysis, Analysis, Dover, 1984.
Theodore W. Gamelin, Complex Analysis, Springer, 2001. D. Sarason, Notes on Complex Function Theory , Hindustan Book Agency, 1994 (or) D. Sarason, Complex Function Theory , 2nd edition, AMS, 2007.
Coordinators: Dr. M. Guru Prem Prasad Department of MathematicsIIT Guwahati
11
12
Complex Logarithm function, Branches of multiple valued functions.
Complex power/exponent functions, Branches of
13
Introducing curves, paths and contours, Statement of Jordan Curve theorem, Orientation of closed curves.
14
Line Integrals (contour integral) and its properties. Fundamental theorem of calculus.
15
Cauchy-Goursat Theorem for simply connected domain (Proof using Green’s theorem)
16
Deformation theorem, Cauchy-Goursat Theorem for multiply connected domain.
17
Deformation and homotopy, Winding number of closed curve and its properties, Homotopy version of Cauchy’s Theorem.
18
Cauchy’s integral formula, Cauchy’s estimate, Liouville’s theorem, Fundamental Theorem of Al gebra.
19
Existence of derivatives of analytic functions, Morera’s theorem.
20
Open mapping theorem, Maximum-Modulus theorem, MinimumModulus theorem.
21
Power series, Power series represents Taylor’s theorem.
22
Analytic continuation by power series, Schwarz Reflection principle.
23
Zeros of analytic functions, Hurwitz theorem.
24
Singularities and its classifications, Behaviour of the function in the neighborhood of singularities.
25
Laurent’s theorem, Finding Laurent Series.
26
Casorati-Weierstrass theorem. Statement of Great Picard Theorem.
27
Argument Principle and Rouche’s theorem.
28
G
Lu
s the
, Residu
t isolated si
analytic function,
ula
oints
29
Residue theorem
30
Evaluating definite integrals of rational functions involving trigonometric functions by converting into contour integration on the unit circle.
31
Evaluating improper integrals of rational functions.
32
Jordan’s lemma, Evaluating improper integrals using Jordan’s lemma.
33
Evaluating integrals of functions with finitely many real poles (Indented contour integration)
34
Evaluating integrals involving branches of multiple valued functions (Keyhole contour i ntegration)
35
Definition of Laplace transform and inversion. Finding inverse Laplace transform using Residue’s theorem.
36
Bilinear transformations and its properties.
37
Conformal mappings and its properties. Statement of Riemann Mapping Theorem.
38
Schwarz-Christoffel transformation, Joukowski transformation.
39
Harmonic functions, Maximum and Minimum principle. Poisson integral formula.
40
Dirichlet problem for a Disk, Dirichlet problem for a Half plane, Neumann problems.
References: 1. J. W. Brown and R. V. Churchill, Complex Variables and Applications, 7th edition, McGraw Hill, 2003. 2. J. H. Mathews and R. W. Howell , Complex Analysis for Mathematics and Engineering, 3rd edition, Narosa, 1998. 3. E. B. Saff and A. D. Snider, Fundamentals of Complex Analysis with Applications to Engineering, Science and Mathematics, 3rd Edition, Prentice Hall, 2002. 4. T. Needham, Visual Complex Analysis, Oxford, 1997. 5. H. A. Priestley, Introduction to Complex Analysis, 2nd Edition (Indian), Oxford, 2006. 6. S. D. Fisher, Complex Variables, 2nd Edition, Dover, 1999. A joint venture by IISc and IITs, funded by MHRD, Govt of India
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