Basic Complex Analysis

Cauchy Integral Theorem, Consequences of the Cauchy Integral Theorem (including holomorphic iff analytic, Local Behavior, Phragmén–Lindelöf, Reflection Principle, Calculation of Integrals), Montel, Vitali, and Hurwitz’s Theorems, Fractional Linear Transformations, Conformal Maps, Zeros and Product Formulae, Elliptic Functions, Global Analytic Functions, and Picard’s Theorem.

Se...

Cauchy Integral Theorem, Consequences of the Cauchy Integral Theorem (including holomorphic iff analytic, Local Behavior, Phragmén–Lindelöf, Reflection Principle, Calculation of Integrals), Montel, Vitali, and Hurwitz’s Theorems, Fractional Linear Transformations, Conformal Maps, Zeros and Product Formulae, Elliptic Functions, Global Analytic Functions, and Picard’s Theorem.

Selected topics include the Goursat Argument, Ultimate and Ultra Cauchy Integral Formulas, Runge’s Theorem, complex interpolation, Marty’s Theorem, continued fraction analysis of real numbers, Riemann mapping theorem, Uniformization theorem (modulo results from Part 3), Mittag–Leffler and Weirstrass product theorems, finite order and Hadamard product formula, Gamma function, Euler–Maclaurin Series and Stirling’s formula to all orders, Jensen’s formula and Blaschke products, Weierstrass and Jacobi elliptic functions, Jacobi theta functions, Paley–Wiener theorems, Hartog’s phenomenon, and Poincaré’s theorem that in higher complex dimensions, the ball and polydisk are not conformally equivalent.

Barry Simon is currently an IBM Professor of Mathematics and Theoretical Physics at the California Institute of Technology. He graduated from Princeton University with his Ph.D in Physics. In 2012 Simon won the International Association of Mathematical Physics’ Poincaré Prize for outstanding contributions to mathematical physics. He has authored more than 400 publications on...

Barry Simon is currently an IBM Professor of Mathematics and Theoretical Physics at the California Institute of Technology. He graduated from Princeton University with his Ph.D in Physics. In 2012 Simon won the International Association of Mathematical Physics’ Poincaré Prize for outstanding contributions to mathematical physics. He has authored more than 400 publications on mathematics and physics.