Syllabus: The algebra and geometry of complex numbers, representations of a complex number. Exponential and logarithmic functions. Diﬀerentiation, analytic functions, Cauchy-Riemann equations. Contour integrals, Independence of path. Cauchy’s Integral Theorem, Cauchy’s Integral Formula, Liouville’s Theorem and its applications. Complex power series, uniform convergence. Removable and isolated singularities, Taylor’s and Laurent’s Theorems. The residue theorem and applications.
Prerequisites: Calculus, Multivariable calculus, Real analysis, Linear algebra.
- L. V. Ahlfors: Complex Analysis, Mcgraw Hill, 1979.
- J. B. Conway: Functions of one complex variable, Springer international students edition.
- T. W. Gamelin: Complex Analysis, Springer, 2003.