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By Monika Malhotra, Prof. Amber Srivastava, Prof. K C Sarangi, Prof. Rohit Mukherjee, Vivek Sharma

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- 978-81-88870-79-0
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4EC2-01: Advance Engineering Mathematics-II

**Unit-1 Introduction:**Objective, scope and outcome of the course.**Unit-2 Complex Variable – Differentiation:**Differentiation, Cauchy-Riemann equations, analytic functions, harmonic functions, finding harmonic conjugate; elementary analytic functions (exponential, trigonometric, logarithm) and their properties; Conformal mappings, Mobius transformations and their properties.**Unit-3 Complex Variable - Integration:**Contour integrals, Cauchy-Goursat theorem (without proof), Cauchy Integral formula (without proof), Liouville’s theorem and Maximum- Modulus theorem (without proof); Taylor’s series, zeros of analytic functions, singularities, Laurent’s series; Residues, Cauchy Residue theorem (without proof).**Unit-4 Applications of complex integration by residues:**Evaluation of definite integral involving sine and cosine. Evaluation of certain improper integrals.**Unit-5 Special Functions**: Legendre’s function, Rodrigues formula, generating function, Simple recurrence relations, orthogonal property. Bessel’s functions of first and second kind, generating function, simple recurrence relations, orthogonal property. Unit-6 Linear Algebra: Vector Spaces, subspaces, Linear independence, basis and dimension, Inner product spaces, Orthogonality, Gram Schmidt orthogonalization, characteristic polynomial, minimal polynomial, positive definite matrices and canonical forms, QR decomposition.