## PHY 230: Mathematics for Physics

3 credits | Prerequisites: MAT 104

### Course rationale

This course is mandatory for the students who wish to have a minor in physics. Mathematics is ubiquitous in physics. Therefore, to understand physics properly it is imperative to have a good mastery in mathematics. This course is designed to abridge any gap in mathematics to understand physics.

### Course content

Complex Analysis: introduction, Cartesian and Polar representation, Euler formula, differentiability, Cauchy-Riemann equation, contour integration, Cauchy-Goursat theorem, Cauchy integral formula, Taylor and Laurent series, Cauchy residue theorem. 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. Vector Analysis: derivatives, and integrals of vector valued functions, gradient vectors and tangent planes; Multivariable Calculus: partial differentiation, functions of several variables, limits and continuity, partial derivatives, vector fields, gradient, divergence, and curl, line, double, and multiple integrals, change of variable in multiple integrals, Jacobian, Green’s theorem, Stokes theorem, Gauss’s theorem. 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 first and second kind, generating function, associated Legendre polynomial, spherical harmonics.

### Course objectives

1. Understand different topics of this course.
2. Apply Fourier analysis to solve differential equations.
3. Understand the basics of complex analysis, and theorems of complex integration.
4. Understand some important special functions of physics.
5. Apply these special functions to different problems arising in physics.

### References

1. Fourier Series and Boundary Value Problems (8th edition) by James Ward Brown, Ruel V. Churchill
2. Fourier Analysis: An Introduction by Elias M. Stein, and Rami Shakarchi
3. Complex Variables and Applications (9th edition) by James Ward Brown, Ruel V. Churchill
4. Special Functions: A Graduate Text by Richard Beals, and Roderick Wong
5. Special Functions by George E. Andrews, Richard Askey, and Ranjan Roy
6. Mathematical Methods for Physicists (7th edition) by George B. Arfken, Hans J. Weber, and Frank E. Harris