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 Bessel's Differential Equation In the Sturm-Liouville Boundary Value Problem, there is an important special case called Bessel's Differential Equation which arises in numerous problems, especially in polar and cylindrical coordinates. Bessel's Differential Equation is defined as: where is a non-negative real number. The solutions of this equation are called Bessel Functions of order . Although the order can be any real number, the scope of this section is limited to non-negative integers, i.e., , unless specified otherwise. Since Bessel's differential equation is a second order ordinary differential equation, two sets of functions, the Bessel function of the first kind and the Bessel function of the second kind (also known as the Weber Function) , are needed to form the general solution: However, is divergent at . The associated coefficient is forced to be zero to obtain a physically meaningful result when there is no source or sink at .
Important Properties

Basic Relationship: The Bessel function of the first kind of order can be expressed as a series of gamma functions.

The Bessel function of the second kind of order can be expressed in terms of the Bessel function of the first kind.

Generating Function: The generating function of the Bessel Function of the first kind is

 Recurrence Relation: A Bessel function of higher order can be expressed by Bessel functions of lower orders.

Asymptotic Approximations: Keeping the first few terms in the series expansions, the behavior of a Bessel function at small or large , can be captured and expressed as elementary functions which are much easier to be understood and calculated than the more abstract symbols and .

For small , i.e., fixed and ,

For large , i.e., fixed and ,

Orthogonality: Suppose that is the non-negative root of the characteristic equation

associated with a physical problem defined on the interval of . It can be verified that

By using this orthogonality, the component of the general solution of the physical problem is

The general solution thus yields

where

and is a piecewise continuous function, generally the non-homogeneous term of the problem.

This orthogonal series expansion is also known as a Fourier-Bessel Series expansion or a Generalized Fourier Series expansion. The transform based on this relationship is called a Hankel Transform.

Hankel Function: Similar to , the Hankel function of the first kind and second kind, prominent in the theory of wave propagation, are defined as

For large , i.e., fixed and ,

Modified Bessel Function: Similar to the relations between the trigonometric functions and the hyperbolic trigonometric functions,

The modified Bessel functions of the first and second kind of order are defined as

and

Glossary