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Advanced mathematical Physics

Introduction Complex Numbers ,Vector Analysis Solution of second order differential

equation with constant coefficients, power series solutions Frobenius method.

Generating function for Hermite polynomials. Hermite differential equation and

polynomials, Integral formula for Hermite polynomial, recurrence formula, Rodrigues

formula, orthogonality of Hermite polynomials. Laguerre differential equations and

polynomials, Generating function for Laguerre polynomials, recurrence relation, Rodriguesformula for Laguerre polynomials, orthogonality property. Beta and gamma functions:

symmetry property, evaluation and transformation of Beta function, evaluation of gamma

function, transformation of gamma function, relation between beta and gamma functions.

Evaluation of integrals using Beta & gamma functions. Hypergeometric equation,

Hypergeometric function: Differentiation of hypergeometric function and its integral

representation , Linear transformations, Representation of various functions in terms of

hypergeometric functions, confluent hypergeometric functions, representation of various

functions in terms of hypergeometric functions. Integral transforms, Fourier transforms

and their properties, convolution theorem for Fourier transforms, Parsevals theorem,

simple applications of Fourier transforms. Evaluation of integrals, solution of boundary

value problems. Laplace transforms and their properties, Laplace transform of derivatives

and integrals. Laplace transform of periodic functions, initial and final value theorem,

Laplace transform of some special functions, inverse Laplace transforms, Convolution

theorem

Text and Reference Books: Mathematical Methods for Physics by George B. Arfken

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MATHEMATICAL PHYSICS

Part-I Co-ordinate System and Vectors

The Co-ordinate system : Orthogonal and non-orthogonal, Right handed and Left Handed Cartesian system,

Necessity for curvilinear co-ordinate system. Polar, cylindrical and general curvilinear coordinate system.

Length, area and volume elements in all these systems.

Vectors : Vector triple product, polar and axial vectors and their examples from Physics. Vectorial equation of

straight line, plane and circle. Base vector, vector transformations. Scalar and vector fields with illustrations from

Physics, directional derivative of a scalar field, gradient of a scalar field, divergence and curl of vector fields and

their physical meaning, Expression for gradient, divergence and curl in curvilinear co-ordinates. Integration,

important identities. Gauss, Green and Stokes theorems and their applications.

Part-II Matrices , special functions and Fourier series

Matrices : Addition law of matrices, matrix multiplication, properties of matrices, special square matrices,

inverse of matrices, Elementary transformation of matrices similarity, orthogonal and unitary transformation.

Eigen value, Eigen vector. Solution of simultaneous linear equations. Diagonalisation of matrix.

Special functions: Beta and Gamma functions, relation between them, recurrence relation for

gamma function.

Fourier Series: Fouriers Theorem

Part-III Differential Equations

Differential equations and special functions : Ordinary differential equations, Homogeneous equations,

solutions in power series, series solution of second order, differential equation by the Froebenius method.

Legendres differential equation, Legendre polynomial, Rodrigues formula, Generating function of Legendre

polynomial, orthogonal properties of Legendre polynomial, Recurrence relation for Pn(X). Bessels Differential

Equation and its solution, Bessels function of first kind. Recurrence relation, spherical Bessels function,

Generating function in connection with Bessells function.

Part-IV Complex Numbers

Complex Variables:- Introduction to complex numbers. Algebra of complex numbers. Argand diagram, algebra

of complex numbers using Argand diagram. Rectangular, polar and exponential forms of complex numbers. De-

Moivres Theorem (statement only). Trigonometric, hyperbolic and exponential functions. Powers, roots and log

of complex numbers. Applications of complex numbers to determine velocity and acceleration in curved motion.Cauchy-Rieman conditions and their applications.