PH5014: Mathematical Methods in Physics

L-T-P-C: 3-0-0-3

Prerequisites: None

Syllabus

Module 1: Complex variables

( Lecture hours: 10 , Tutorial hours: 0 , Lab hours: 0 )

Analytic functions of a complex variable, differentiation and the Cauchy-Riemann conditions, Contour integrals and Cauchy's theorem, Taylor and Laurent series, Singularities: poles, essential singularities, residue theorem, Analytic continuation, multiple-valued functions, branch points and branch cut integration .

Module 2: Integral Transforms

( Lecture hours: 6 , Tutorial hours: 0 , Lab hours: 0 )

Sturm-Liouville problems, Expansion in eigenfunctions, Fourier series and integrals, Fourier and Laplace transforms .

Module 3: Differential Equations and Special functions

( Lecture hours: 15 , Tutorial hours: 0 , Lab hours: 0 )

ODEs: Classification, Basic solution methods, Exact differentials, Integrating factors, Series solution, singular points, Special Functions: Bessel functions, Legendre polynomials, Spherical harmonics, Gamma function, PDEs: Classification of PDEs, boundary conditions. Characteristics. Separation of variables. Solution using integral transforms .

Module 4: Linear Algebra

( Lecture hours: 6 , Tutorial hours: 0 , Lab hours: 0 )

Matrices: Rotations and orthonormal matrices, Diagonalization of Matrices, Tensor Analysis, Contraction, Direct Product, Transformation of tensors under rotation .

Module 5: Probability theory

( Lecture hours: 5 , Tutorial hours: 0 , Lab hours: 0 )

Random variables: basic notions of probability, distributions, moments and cumulants, generating functions, discrete and continuous distributions, The central limit theorem .

Study Materials

Textbooks

  1. Mathematical Methods for Physicists, G. Arfken and H. J. Weber, Academic Press, 6th Edition, Indian Edition (2005), ISBN: 978-9381269558
  2. Mathematical Physics, Vol I, S. D. Joglekar, University Press (2005), ISBN: 978-8173714221
  3. Mathematical Physics, Vol II, S. D. Joglekar, University Press, ISBN: 978-8173715600
  4. Advanced Engineering Mathematics, E. Kreyszig, 9th Edition, ISBN: 978-8126531356
  5. Numerical Methods for Engineers, Steven Chapra, Raymond Canale, 6th Edition, ISBN: 978-0-07-340106-5
  6. Numerical Methods for Scientific and Engineering Computation, M. K. Jain, S. R. K. Iyengar, R. K. Jain, New Age International (2003), ISBN: 978-8122414615

References

  1. Complex Variables and Applications, J. W. Brown, R. V. Churchill, McGraw-Hill Education, ISBN: 978-0073383170
  2. Mathematics for Physicists, P. Dennery and A. Krzywicki, Dover Publications, ISBN: 978-0486691930
  3. Mathematical Physics: With Applications, Problems and Solutions, V. Balakrishnan, Ane Books (2017), ISBN: 9386761114