## PHY 224: Complex Analysis

3 credits | Prerequisites: PHY 220

### Course rationale

The beautiful discovery of imaginary numbers is a great leap in mathematics. The real numbers get extended to complex numbers. This course elucidates the algebra and calculus of complex variables. The application of complex analysis is diverse. In physics, the complex variable is manifest almost in everywhere from optics to quantum mechanics, to contemporary string theory. This course will form the basis to enter these applications of complex variables in physics

### Course content

Introduction to Complex Numbers: complex plane, algebra of complex numbers; Functions of a Complex
Variable: mapping, limit, continuity, differentiability of a complex function, bilinear transformations; Analytic Functions: Cauchy-Riemann equations, necessary and sufficient conditions for differentiability; Harmonic Functions: Laplace equation in 2-dimensions in Cartesian and polar coordinates, relation with analytic function; Power Series: Taylor’s and Laurent’s expansion; Complex Integration: line integration over rectifiable curves, winding number, Cauchy’s Theorem: Cauchy’s integral formula, Liouville’s theorem, fundamental theorem of algebra; Residue Calculus: singularities, residues, Cauchy’s residue theorem, evaluation of integrals by contour integration, Rouche’s theorem; Conformal Mappings: preservation of angle, scale factors, applications; Meromorphic Functions: infinite product decomposition of entire functions; Schwarz-Christoffel Transformation: mapping real axis onto a polygon, triangles, and rectangles; Analytic Continuation: monodromy theorem.

### Course objectives

1. Analyze and understand the functions of a complex variable.
2. Use analytic functions to solve the Laplace equation.
3. Apply Cauchy’s theorem to evaluate improper integrals.
4. Use residue calculus to evaluate different integrals in physics and mathematics.
5. Understand and apply different theorems and transformations of complex analysis to physics.

### References

1. Complex Variables and Applications (8th edition) by James Ward Brown, Ruel V. Churchill
2. Complex Variables: Introduction and Applications (2nd edition) by Mark J. Ablowitz, and Athanassios S. Fokas
3. Complex Analysis (3rd edition) by Joseph Bak, and Donald J. Newman
4. Complex Analysis: A first Course with Applications by Dennis G. Zill, and Patrick D. Shanahan
5. Complex Analysis by Elias M. Stein, and Rami Shakarchi