Bessel functions

Bessel functions arise in the study of the Laplace operator and related operators on domains for which it is convenient to use polar coordinates. Here is a set of notes on Bessel functions, prepared for my differential equations class, Math 524. Bessel Functions

The following gives an alternative treatment of Bessel functions. It is briefer, but it also discusses the Hankel transform

The following brief notes establish two important integral formulas involving Bessel functions, the Lipschitz-Hankel integral formula and the Weber integral formula, of use in the applications of Bessel functions to PDE on cones.

Airy functions

Airy functions arise in diffraction theory. These notes are excerpted from an appendix in my monograph with R. Melrose, “Boundary problems for wave equations with grazing and gliding rays.” Airy functions and Airy quotients

Figures used in these Airy function notes can be found on pages 6-10 of the following file. Figures

Riemann zeta function

The Riemann zeta function conveys a lot of information about the set of prime numbers. The following notes discuss this. The zeta function and the prime number theorem

Background material on the zeta function, of particular use in Appendix A of the notes above, can be found here. This is excerpted from my “Introduction to Complex Analysis.”

The gamma function

The gamma function is the most basic of the “higher transcendental functions.” It makes an appearance in formulas for Bessel functions, Airy functions, the Riemann zeta function, and other important special functions in analysis.

The following notes treat this function. The gamma function. This is excerpted from my “Introduction to Complex Analysis.”The asymptotic behavior of Gamma(z) for large z is given by Stirling’s formula, treated in the following note.  Stirling’s formula and the Schrodinger equation.

The following is a speculative account of How Euler might have come up with the gamma function

Elliptic functions

Elliptic functions are an intriguing class of doubly periodic meromorphic functions on the complex plane. The following notes are excerpted from my “Introduction to Complex Analysis.” Elliptic functions

The following notes treat the Weierstrass P-function as a principal-value distribution on a torus, and compute its Fourier series, as well as the Fourier series of other elliptic functions. The Weierstrass P-function and its Fourier series

The Fresnel integral

The Fresnel integral, related to the error function via analytic continuation, arises in studies of diffraction. It also has close connections to the study of 1D Schrodinger equations, discussed in these notes. The Schrodinger equation and the Fresnel integral