15.1 The Bessel Function

The Bessel functions, denoted as J _n( x), are used in engineering, acoustics, aeronautics, thermodynamics, theory of elasticity and others. For instance, in the electrical engineering field, they are used in frequency modulation, transmission lines, and telephone equations.

Bessel functions are solutions of the differential equation

where n can be any number, positive or negative integer, fractional, or even a complex number. Then, the form of the general solution of (15.1) depends on the value of n.

Differential equations with variable coefficients, such as (15.1), cannot be solved in terms of familiar functions as those which we encountered in ordinary differential equations with constant coefficients. The usual procedure is to derive solutions in the form of infinite series, and the most common are the Method of Frobenius and the Method of Picard. It is beyond the scope of this book to derive the infinite series which are approximations to the solutions of these differential equations; these are discussed in advanced mathematics textbooks. Therefore, we will accept the solutions without proof.

Applying the method of Frobenius to (15.1), we obtain the infinite power series

This series is referred to as Bessel function of order n where n is any...