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### Overview

This course is a basic course offered to UG/PG students of Engineering/Science background. It contains Analytic Functions, applications to the problems of potential flow, Harmonic functions, Harmonic conjugates, Milne’s method, Complex integration, sequences and series, uniform convergence, power series, Hadamard’s formula for the radius of convergence, Taylor and Laurent series, zeros and poles of a function, meromorphic function, the residue at a singularity, Residue theorem, the argument principle and Rouche’s theorem, contour integration and its applications to evaluation of a real integral, integration through a branch cut, conformal mapping, application to potential theory, review of unilateral and bilateral Z-transforms and their properties, application of calculus of residues for the inversion formula of Z- transforms and Laplace transforms, review of Fourier integrals and Fourier transforms, Finite Fourier transforms, discrete Fourier transforms and applications, basic concepts of probability, Bayes theorem, probability networks, discrete and continuous probability distribution, joint distribution, correlation coefficient, applications to problems of reliability, queueing theory, service time for a customer in a facility and life testing, testing of hypotheses. This course has tremendous applications in diverse fields of Engineering and Sciences such as Signal processing, Potential theory, Bending of beams etc.INTENDED AUDIENCE: UG and PG students of technical institutions/ universities/colleges.PREREQUISITES : NoneINDUSTRY SUPPORT : None

### Syllabus

Week 1: Analytic Functions, Cauchy-Riemann Equations, Harmonic Functions, Harmonic Conjugates and Milne’s Method, Applications to the problems of potential flow-I, Applications to the problems of potential flow-IIWeek 2 : Complex integration, Cauchy’s theorem-I, Cauchy’s theorem-II , Cauchy’s Integral Formula for the Derivatives of an Analytic Function , Morera’s theorem, Liouville’s theorem and Fundamental Theorem of AlgebraWeek 3 : Winding Number and Maximum Modulus Principle, Sequences and Series, Uniform Convergence of Series, Power Series, Taylor seriesWeek 4:Laurent Series,Zeros and Singularities of an Analytic Function,Residue at a Singularity,Residue Theorem,Meromorphic FunctionsWeek 5 : Evaluation of real integrals using residues-I, Evaluation of real integrals using residues-II , Evaluation of real integrals using residues-III, Evaluation of real integrals using residues-IV, Evaluation of real integrals using residues-VWeek 6 : Bilinear Transformations, Cross ratio, Conformal Mapping-I, Conformal Mapping-II, Conformal mappings from half plane to disk and half plane to half plane-IWeek 7: Conformal mappings from disk to disk and angular region to disk, Application of Conformal mapping to potential theory, Review of Z-transforms-I, Review of Z-transforms-II, Review of Z-transforms-IIIWeek 8: Review of bilateral Z-transforms, Finite Fourier transforms, Fourier integrals and Fourier transforms, Fourier Series, Discrete Fourier transforms-IWeek 9: Discrete Fourier transforms-II, Basic concepts of probability, Conditional probability, Bayes theorem and Probability networks, Discrete probability distributionWeek 10: Binomial distribution, Negative binomial distribution and Poisson distribution, Continuous probability distribution, Poisson Process, Exponential distributionWeek 11: Normal distribution , Joint distribution-I, Joint probability distribution-III, Joint probability distribution-III, Correlation and regression-IWeek 12: Correlation and regression-II, Testing of hypotheses-I, Testing of hypotheses-II, Testing of hypotheses-III, Application to Queueing Theory and Reliablility Theory

### Taught by

Prof. P.N. Agarwal