Mumbai University
APPLIED MATHEMATICS III
APPLIED MATHEMATICS III
Complex Variables: Functions of complex variables, continuity(only statement), derivability of a function, analytical regular function, necessary condition for a function to be analytic, statement of sufficient conditions, Cauchy Riemann equations in polar co-ordinates. Harmonic functions, orthogonal trajectories, Analytical and Milne Thomson method to find f(z) from its real or imaginary part. Mapping- conformal mapping, linear and bilinear mapping with geometrical interpretations.
Fourier Series and Integrals: Orthogonal and orthonormal functions, expression of a function in a series of orthogonal functions, sine and cosine functions and their orthogonality properties. Fourier series, Drichlet conditions, periodic functions, even and odd functions, half range sine and cosine series, Parseval’s relation. Complex form of Fourier series, introduction to Fourier integral, relation with Laplace transform.
Laplace Transforms: Function of bounded variable ( statement only ), Laplace transforms of 1, at, exp( at ), sin( at ), cos( at ), sinh(at), cosh(at), erf(t), shifting properties, expressions with proofs for L { t f(t) }, L { f(t)/t }, Laplace of an integral and derivative. Unit step functions, Heavyside, Dirac Delta functions and their Laplace transform, Laplace transform of periodic functions. Evaluation of inverse Laplace transforms, partial fraction method, Heavyside development, Convolution theorem. Application to solve initial and boundary value problems involving ordinary differential equations with one variable.
Matrices: Types of matrices, adjoint of a matrix, inverse of a matrix, elementary transformations, rank of a matrix, linear dependent and independent rows and columns of a matrix over a real field, reduction to a normal form, partitioning of matrices System of homogenous and non homogenous equations, their consistency and their solutions.
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