Global Cauchy Theorem

Luis T. Magalhães
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Luis T. Magalhães: University of Lisbon, Instituto Superior Técnico

Chapter Chapter 7 in Complex Analysis and Dynamics in One Variable with Applications, 2025, pp 115-141 from Springer

Abstract: Abstract Chains, cycles and homology, global Cauchy theorem and formula, path homotopy and integral invariance, simply and multiply connected regions, homology basis, extensions of the maximum modulus principle including the Phragmén–Lindelöf Principle, Hardy Uncertainty Principle, order and type of entire functions, Denjoy–Carleman–Ahlfors theorem on the number of asymptotic values of entire functions. Exercises, including on Laplace transform and its application to solve differential equations, interpolation theorem for entire functions known as Sampling Theorem in telecommunications and related to the Nyquist sampling frequency, the Hadamard 3 circles theorem, and with applications to linear systems analysis and control, including the notions of transfer function, impulse response, Bode diagram, and linear feedback control systems.

Date: 2025
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DOI: 10.1007/978-3-031-64999-8_7

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