PHY 401 Mathematical Methods of Optics and Physics

Semester: Fall
Credit Hours: 4
Prerequisite: MTH 164, 282, or equivalent

Study of mathematical techniques such as contour integration, transform theory, Fourier transforms, asymptotic expansions, and Green's functions, as applied to differential, difference, and integral equations. (Prior Titles: Complex Analysis and Diff Equations & Mathematical Methods of Theoretical Optics). (Cross-listed with OPT411).

syllabus

1. The field of complex numbers; norm of a complex number; complex plane; stereographic map of the sphere.
2. Closed an open sets; convergent sequences; infinite series; absolute convergence; double series. Infinite products.
3. Definition of a continuous function of a complex variable; definition of an analytic function and a regular function; Cauchy-Riemann equation; harmonic functions.
4. Polynomials; power series; exponential and related functions.
5. Countour integrals; Cauchy's theorem;' Taylor's theorem, Laurent's theorem. Residue calculus; Principal value; Hilbert transform.
6. Entire functions; product representation of trigonometric functions. Weierstrass functions. The Gamma function. Product formula. Asymptotic formula.
7. Conformal mapping; solution of the Laplace equation; Dirichlet problem.
8. Divergent series. Borel summation; Pade approximants
9. Linear ordinary differential equations; ordinary point, regular and irregular singular points; the point at infinity; local behavior near regular singular points.
10. Solution near an ordinary point; wronskian; solution near a regular singular point.
11. The hypergeometric equation. Solution by power series. Integral representation. Behavior at infinity.
12. Bessel functions; Legendre functions; Hermite's equation. Sturm-Liouville problems.
13. Local behavior near irregular singular points; asymptotic series.
14. Fourier and Laplace transforms; solution of deferential equations by Fourier and Laplace transforms
15. Linear second order partial differential equations; canonical form; hyperbolic elliptic and parabolic types; Cauchy problem.

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