Syllabus for the Midterm Exam for Prof. W. Kahan's Math. 185 section #2 289 Cory Hall, 1:15 pm. - 2, 30 Oct. 2006 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The exam's subject matter includes everything in chapters I to VII incl. of the course text "Notes on Complex Function Theory" by Donald Sarason (1994, Henry Helson, Berkeley) plus the following notes posted on the class web page : 30Aug06S.pdf Cnfrml.pdf Conics.pdf DSpr2p43.pdf Derivative.pdf Errata.pdf Ex19.pdf Mobius.pdf except for text printed in small (10-pt.) type. Here is a list of topics covered in the course so far: ------------------------------------------------------ Cauchy-Riemann equations Derivatives as limits or confluent divided differences Inverse functions and implicit differentiation Conformal maps, Critical points "Conformal" <=> "Holomorphic" => "Harmonic" Harmonic functions and their conjugates Recovery of harmonic conjugates by integration of derivatives Recovery of harmonic conjugates without integration Maximum of harmonic functions occurs on a boundary (Ill-)determination of a harmonic function by its values on an uncountable set of points. Branch-points and slitted domains of Principal Values Elementary transcendental functions and their inverses Conformal maps of circles and lines to conic sections Bilinear-Rational / Linear Fractional / Mšbius Functions Clircles to clircles Isomorphic to Quotient Group of 2-by-2 matrices ... Characterization by its fixed-points Preservation of Cross Ratios Stereographic Projection and the Riemann Sphere Chordal distance Inversion/Reflection in a circle or line Series Absolute Convergence, Uniform Convergence Power series, radius of convergence Algebra (sums, products, quotients, ...) of series Term-by-term derivatives and integrals Integrals of complex functions Length of a curve Integrals of Complex Analytic Functions Independence of path of integration Cauchy integrals are holomorphic Cauchy's Integral Formula for an Analytic Function's value inside a closed curve in terms of its values on the curve Liouville's Theorem about bounded entire functions Gauss's Theorem that polynomials have zeros Analytic functions' zeros are isolated Analytic functions are (ill-)determined by their values on an uncountable point-set. Uniform convergence of analytic functions to another Maximum modulus principle, D'Alembert's principle Schwarz's Lemma bounds the magnitude of an analytic function ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The midterm exam will be a closed-book exam to which you must bring no notes, texts, papers, calculating nor communicating instruments like cell-phones. Blank paper will be supplied for both scratch use and for submission of material to be graded, so bring no paper of your own. The washrooms on Cory Hall's second floor are closed for renovation, so do whatever need be done to preclude visits to a washroom or water fountain during the exam's 45 min. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~