Notes on various topics in mathematics and theoretical physics.

Main sources:

• Stein and Shakarchi, Complex Analysis
• Shabat, Introduction to Complex Analysis
• Ahlfors, Complex Analysis
• Problems and exercises:
  • Shakarchi, Problems and Solutions for Complex Analysis
  • Cahill, Physical Mathematics
  • Hassani, Mathematical Physics
  • Arfken, Harris, and Weber, Mathematical Methods for Physicists

Table of contents:

1. Preliminaries to Complex Analysis
   • 1.1. Complex numbers
   • 1.2. Topology of C
   • 1.3. Functions on C
   • 1.4. Holomorphicity
   • 1.5. Infinite series
   • 1.6. Integration along curves
2. Cauchy's theorem and its applications
   • 2.1. Cauchy's theorem
   • 2.2. Miracles of complex analysis
   • 2.3. Further applications
3. Singularities, residues and meromorphic functions
   • 3.1. The Laurent series
   • 3.2. Singularities
   • 3.3. Classification of holomorphic functions
   • 3.4. Residues
   • 3.5. The complex logarithm
   • 3.6. The argument principle and its applications
   • 3.7. Winding numbers
4. Exercises
   • 4.1. Complex functions
   • 4.2. Limits and power series
   • 4.3. The Laurent expansion
   • 4.4. Residues
   • 4.5. Complex integration
   • 4.6. Evaluation of definite integrals
     • 4.6.1. Real integrals
     • 4.6.2. Jordan's lemma
     • 4.6.3. Singularities on a contour
     • 4.6.4. Multiple singularities on a contour
     • 4.6.5. Avoiding branch cuts

Main sources:

• Szekeres, A Course in Modern Mathematical Physics
• wiki

Table of contents:

1. Sets and structures
   • 1.1. Naive set theory
   • 1.2. Relations
   • 1.3. Mappings
   • 1.4. Infinite sets
   • 1.5. Physics
   • 1.6. Category theory
2. Measure theory and integration
   • 5.1. Measurable spaces
   • 5.2. Measurable functions
   • 5.3. Measure spaces
   • 5.4. Lebesgue measure
   • 5.5. Lebesgue integration
3. Distributions
   • 6.1. Test functions
   • 6.2. Distributions
   • 6.3. Operations on distributions
   • 6.4. Change of variable in δ-function
   • 6.5. Fourier transform
   • 6.6. Green's function
4. Exercises
   • 8.1. Sets and mappings
   • 8.2. Distributions
   • 8.3. Fourier transforms
   • 8.4. Green's function