# Mathematical Methods of Physics

### PHYSICS760

Includes topics in probability theory, complex analysis, asymptotic expansions, group theory, Fourier analysis, Green functions, ordinary and partial differential equations; and use of Mathematica. 3 units.

#### Possible principal texts:

1. K.F. Riley, M.P. Hobson and S.J. Bence Mathematical Methods for Physics and Engineering.
2. D.A. McQuarrie Mathematical Methods for Scientists and Engineers.

#### Other texts to consider:

1. G.B. Arfken and H.J. Weber Mathematical Methods for Physicists.
2. J.J. Kelly Graduate Mathematical Physics.

#### Prerequisites:

Undergraduate courses in intermediate calculus (such as MTH 103 or equivalent) are required. Students must have some familiarity with partial differentiation, multiple integrals, infinite series, differential vector calculus (grad, div, curl, Laplacian), integral vector calculus (divergence and Stokes theorems), coordinates (spherical and cylindrical), matrices and determinants, simultaneous linear equations, Fourier series, and complex numbers.

#### Syllabus

• Probability: distributions, generating functions, central limit theorem, stochastic processes.
• Complex Variables: analytic functions, complex integrals, residues, contour integration asymptotic expansions.
• Group Theory: definitions, examples, applications, representations, characters, product reps, Clebsch-Gordan coefficients, irreducible representations, irreducible Tensors, Wigner-Eckart.
• Fourier transforms, delta functions, convolution, correlation, power spectrum density.
• ODEs: exact solutions, series solutions, Legendre polynomials and functions, Frobenius method, Bessel functions, eigenfunctions, orthogonal functions, Sturm-Liouville theory, Green's functions, qualitative methods, numerical methods.
• PDEs: separation of variables, cylindrical coordinates (Bessel), spherical coordinates (Legendre and Spherical Harmonics), Green's functions, boundary problems.

### Sample lecture schedule

(based on 25 lectures each of duration 75 minutes)

• Lecture 1: Probability 1 – Probability Distributions
• Lecture 2: Probability 2 – Generating Functions and the Central Limit Theorem
• Lecture 3: Probability 3 – Stochastic Processes
• Lecture 4: Complex Variables 1 – Analytic Functions
• Lecture 5: Complex Variables 2 – Complex Integrals
• Lecture 6: Complex Variables 3 – Residues and Contour Integration
• Lecture 7: Complex Variables 4 – Contour Integration
• Lecture 8: Asymptotic Expansions
• Lecture 9: Group Theory 1 – Definitions, examples, applications
• Lecture 10: Group Theory 2 – Representations and Properties
• Lecture 11: Group Theory 3 – Characters, Product Reps and Clebsch-Gordan coefficients
• Lecture 12: Group Theory 4 – Irreducible Representations and Irreducible Tensors, Wigner-Eckart
• Lecture 13: Fourier Transforms and Delta Functions
• Lecture 14: Convolution, Correlation, and Power Spectrum Density
• Lecture 15: ODEs 1 – Exact Solutions
• Lecture 16: ODEs 2 – Series Solutions – Legendre polynomials and functions
• Lecture 17: ODEs 3 – Series Solutions – Frobenius method and Bessel functions
• Lecture 18: ODEs 4 – Qualitative Methods
• Lecture 19: ODEs 5 – Qualitative Methods and Numerical Methods
• Lecture 20: Green's Functions (1D)
• Lecture 21: Eigenfunctions and Orthogonal Functions
• Lecture 22: Sturm-Liouville Theory
• Lecture 23: PDEs 1 – Separation of Variables and Cylindrical Coordinates (Bessel)
• Lecture 24: PDE 2 – Spherical Coordinates (Legendre and Spherical Harmonics)
• Lecture 25: PDE 3– Green's Functions and Boundary Problems in 3D