Date of Award
Bachelor of Science (B.S.) in Mathematics and University Honors
Bessel functions, Differential equations -- Numerical solutions, Differential equations -- Asymptotic theory, Generating functions
This paper is a deep exploration of the project Bessel Functions by Martin Kreh of Pennsylvania State University. We begin with a derivation of the Bessel functions, which are two solutions to Bessel’s differential equation. Next we find the generating function and use it to prove some useful standard results and recurrence relations. We use these recurrence relations to examine the behavior of the Bessel functions at some special values. Then we use contour integration to derive their integral representations, from which we can produce their asymptotic formulae. We also show an alternate method for deriving the first Bessel function using the generating function. Finally, a graph created using Python illustrates the Bessel functions of order 0, 1, 2, 3, and 4.
Deal, Joella Rae, "Basics of Bessel Functions" (2018). University Honors Theses. Paper 546.