4 - Bessel Functions and Confluent Hypergeometric Functions

Summary

In this chapter, we discuss the confluent hypergeometric equation and the related Bessel and Whittaker equations. The Bessel equation is important in mathematical physics because it arises from the Laplace equation when there is cylindrical symmetry. The confluent hypergeometric equation is obtained when we start with a second-order differential equation whose only singularities are regular singularities at 0, b, and ∞; we let b → ∞. The resulting equation has ∞ as an irregular singular point obtained from a confluence of two regular singularities. Thus, the confluent equation can be derived from the hypergeometric equation by changing the independent variable x to x/b and letting b → ∞. The solutions are ₁F₁ functions, and some properties of these functions are limits of properties of ₂F₁ functions. However, it is often easier to derive the results directly than to justify the limiting procedures.

Whittaker transformed the confluent equation to one in which the coefficient of the first derivative is zero. Solutions of this equation are called Whittaker functions. We find their series and integral representations and their asymptotic behavior and then give some important examples such as the error function and the parabolic cylinder function.

The Bessel equation can be derived from a particular Whittaker equation and can be solved to obtain the Bessel functions to which we devote a good portion of this chapter. These functions are also important for their role in Fourier transforms in several variables.