AIS

Advanced Instructional School (AIS-COMPLEX)

Complex Analysis

Funded by
National Board for Higher Mathematics

Jointly Organized by
Bhaskaracharya Pratishthana, Pune and Dept. of Mathematics, University of Pune

5 June-2 July, 2008

Speakers

A. R. Shastri: See the detailed syllabus
Dinesh Thakur: See the detailed syllabus
A. Mangasuli:
1. Covering space theory (3 lectures)
2. De Rham Cohomology and Hodge Theory (An overview) (1 lecture)
Ravi Raghunathan: The convexity principle and applications to functional analysis and number theory.
R. R. Simha: Riemann Surfaces
S. A. Katre: Doubly periodic functions
Kaushal Verma: (6 lectures)
1. Statement of Picard type theorems on the plane.
2. Definition of Fatou-Bieberbach domains, their properties and the theory of normal forms for local holomorphic automorphisms with fixed points.
3. Fatou and Bieberbach's construction.
4. Theorems of Rosay-Rudin for constructing Fatou-Bieberbach type domains.
5. Examples and connections with complex dynamics in higher dimensions.
Ravi Kulkarni:
1. Geometry of Complex Numbers
2. Branched Covering Space Theory
3. The modular Group
S. R. Ghorpade: Hardy-Ramanujan-Rademacher formula for partitions
Ajit Iqbal Singh: Weierstrass' Product Theorem and Mittag-Leffler's Theorem from Complex-valued to the Banach Algebra set-up.
Please refer to the link: The Mittag-Leffler Theorem: The Origin, Evolution, and Reception ...
Sanjay Pant: Complex dynamics.

Detailed Syllabus (to be updated)

1 Contour Integration 1
1.1 Path Connectivity . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Deﬁnition and Basic Properties of Contour Integration . . . . . . . 6
1.3 Existence of Primitives . . . . . . . . . . . . . . . . . . . . . . . 17
1.4 Cauchy-Goursat Theorem . . . . . . . . . . . . . . . . . . . . . . 21
1.5 * Cauchy’s Theorem via Green’s Theorem . . . . . . . . . . . . . . . 27
1.6 Cauchy’s Integral Formulae . . . . . . . . . . . . . . . . . . . . . . . . 30
1.7 Analyticity of Complex Differentiable Functions . . . . . . . . . . . . . 33
1.8 A Global Implication: Liouville . . . . . . . . . . . . . . . . . . . . 37
1.9 Mean Value and Maximum Modulus . . . . . . . . . . . . . . . . . . 39
1.10 Miscellaneous Exercises. . . . . . . . . . . . . . . . . . . . . 41

2 General Form of Cauchy’s Theorem 45
2.1 Winding Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.2 Homotopy and Simple Connectivity . . . . . . . . . . . . . . . . . 51
2.3 Homology Form of Cauchy’s Theorem . . . . . . . . . . . . . . . . 55
2.4 Miscellaneous Exercises . . . . . . . . . . . . . . . 60

3 Convergence in Function Theory 63
3.1 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 The Exponential and Trigonometric Functions . . . . . . . . . . . . . 76
3.3 Sequences of Holomorphic Functions . . . . . . . . . . . . . . . . . 85
3.4 Convergence Theory for Meromorphic Functions . . . . . . . . . . . . . 89
3.5 Partial Fraction Development of π cot πz. . . . . . . . . . . . . . . . . . 94
3.6 Inﬁnite Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.7 Runge’s Approximation Theorem . . . . . . . . . . . . . . . . . . . . . . 104

4 Normal Families and Conformal Mappings 115
4.1 Metric on Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.2 Normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.3 Equicontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.4 Families of Meromorphic Functions . . . . . . . . . . . . . . . . . . . . . 121
4.5 Uniformization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.6 Miscellaneous Exercises . . . . . . . . . . . . . . . . . . . . . . 128

5 Harmonic Functions 129
5.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.2 Application to Potential Theory . . . . . . . . . . . . . . . . . . . . . . . 138
5.3 Mean Value Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.4 Harnack’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.5 Subharmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.6 Perron’s Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.7 Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
5.8 Multi-connected Domains . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.9 Miscellaneous Exercises . . . . . . . . . . . . . . . . . . . . . . 169