Complex Analysis III
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If the complex derivatives of all orders of such a function are zero, then the function is 0.
6 points, SCA Band 2, 0.125 EFTSL
Consequently if the function is not identically 0, then any zero has a nonnegative integer order, and the zeros of the function are isolated. Other consequences are the Maximum Modulus Theorem, a description of the behavior at poles, Weierstrass's result on essential singularities, and the Inverse Function Theorem.
Section B7 examines the exponential function and its local invertibility. This examination leads to the definition of winding number for a closed curve about a point, and the general form of the Cauchy Integral Formula for a disk follows. Section B10 states and proves a first form of the Residue Theorem for evaluating the complex line integral of a function analytic except for poles.
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Section B11 uses the the first form of the Residue Theorem to evaluate a number of examples of real definite integrals. Section B12 extends the Cauchy Integral Theorem from closed curves in disks to cycles in simply connected regions, and it derives a corresponding version of the Residue Theorem. Section B13 examines the extent to which the results of Section B12 extend to general regions when the cycle is assumed to be a boundary cycle. Section B14 develops the Laurent series expansion of a function analytic in an annulus washer.
As a consequence the nature of essential singularities becomes a little clearer.
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Section B15 introduces holomorphic functions of several variables, showing the equivalence of various definitions of such functions. This material is not used until Advanced Real Analysis. Source Anthony W.
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- Department of Mathematics.
- Department of Mathematics at Columbia University - Complex Analysis and Riemann Surfaces;
Instruction Lectures and problem solving sessions. Assessment Written examination at the end of the course. Other directives The course can not be included in higher education qualification together with the course Complex Analysis 1MA , 5 credits.
Syllabus Revisions Latest syllabus applies from week 30, Previous syllabus applies from week 34, Previous syllabus applies from week 32, Previous syllabus applies from week 31, Previous syllabus applies from week 30, Previous syllabus applies from week 44, Previous syllabus applies from week 29, Previous syllabus applies from week 27, Outlines of Pade Approximation. Pages Numerical Methods in Conformal Mapping. On Complex Rational Approximation.
Topics in Computational Complex Analysis: I. Methods of Descent for Polynomial Equations. Fast Fourier Transform Algorithms with Applications. Error Analysis of Complex Arithmetic. Chebyshev Approximation on the Unit Disk.
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Computer Algebra and Complex Analysis. Short Communications. Bultheel, L. Cuyt, J. Della Dora, R. Farwig et al.