## PHY 225: Mathematical Methods for Physicists

3 credits | Prerequisites: PHY 222, 224

### Course rationale

This is a core course for physics major students. Equipped with the other core courses, this course solidifies the mathematical foundation of a physics major student. There are three major topics in this course: 1) Fourier Analysis, 2) Finite Group Theory, and 3) Special Functions. All these topics find numerous appearances in physics., and this course prepares the students to embark on their journey into many areas of physics.

### Course content

Fourier Series: definition, Fourier exponential series, trigonometric Fourier series; Fourier Transform: Integral transform, Fourier integral, inverse Fourier transform, properties of Fourier transform, Perseval’s theorem; Poisson’s summation formula, Fourier transform in Rd; Finite Group: group axiom, the finite group at low order, permutation group, simple groups, Young tableaux; Finite Group Representations: introduction, Schur’s lemma, Kronecker products, real and complex representation; Group Character: definition, projection formula, examples, Frobenius’s character formula; SU(2): introduction, some representations, from Lie algebra to Lie groups, applications; Beta and Gamma Function: beta and gamma integrals, Euler product, reflection formula, duplication formula, asymptotics, digamma function, Hurwitz and Riemann zeta function; Series Solution of Second Order ODEs: Frobenius’s method, series around a regular point, expansion around a regular singular point; Bessel Function: differential equation, generating function, Hankel function, asymptotic expansion; Legendre Polynomial: differential equation, Legendre function of the first and second kind, generating function, associated Legendre polynomial, spherical harmonics; Hypergeometric Function: hypergeometric series, Euler’s integral representation, the hypergeometric equation, contiguous relations, quadratic transformations, transformation and some special values; Sturm-Liouville Theory: orthogonal polynomials, examples, Hermitian operators, generalized Fourier series.

### Course objectives

- Understand the three different topics of this course.
- Apply Fourier analysis to solve differential equations.
- Apply the basic group theory to different problems.
- Understand some important special functions of physics.
- Apply these special functions to different problems arising in physics.

### References

- Fourier Series and Boundary Value Problems (8th edition) by James Ward Brown, Ruel V. Churchill
- Fourier Analysis: An Introduction by Elias M. Stein, and Rami Shakarchi
- Group Theory (A Physicist’s Survey) by Pierre Ramond
- Representation Theory: A First Course by William Fulton, and Joe Harris
- Special Functions: A Graduate Text by Richard Beals, and Roderick Wong
- Special Functions by George E. Andrews, Richard Askey, and Ranjan Roy
- Mathematical Methods for Physicists (7th edition) by George B. Arfken, Hans J. Weber, and Frank E. Harris