Another important class of special functions is the so-called Bessel Functions. These functions are applicable in a wide variety of situations and, similar to the other special functions, one particular set of Bessel functions also has the property of orthogonality. This subsection overviews the definition and development of the Bessel functions and highlights some key features that are useful in practical application.

Bessel’s Equation

The ordinary Bessel equation is given as

where is referred to as the order of the Bessel function and is a parameter within the argument of the resultant Bessel functions. If we let , then

and

Therefore, with these substitutions, eqn. (8.37) becomes

This form, written with t = x, gives

This is the most common representation of Bessel’s equation. This is somewhat unfortunate since eqn. (8.37) is more general and actually occurs more frequently in practice. However, as shown here, the extension to included a parameter is straightforward (we simply replace x with ).

One Solution via the Power Series Method

Since eqn. (8.39) has a regular singular point at x = 0, we need to use the extended power series method. Thus, we try

and, upon substitution of this assumed solution and its derivatives into the defining differential equation, the indicial equation becomes (for ),

Therefore, one gets two roots: and .

Focusing first on , the recurrence relation becomes

for . This form is a little different from usual. In particular, since in the typical representation all the odd coefficients vanish (i.e. ), we simply replaced m with 2m-2, which then reduces to eqn. (8.42), with the index m varying from 1 to in unit increments (this is why the above coefficient is written as a_2m).

Finally, to put the solution into standard form, we define a₀ as

and the first solution to the ordinary Bessel's equation becomes

This function is called an ordinary Bessel function of the first kind and it is denoted by . After some manipulation, the infinite series representation for can be written as

Performing similar operations for gives a second solution to eqn. (8.39), or

This series converges for all values of x except for x = 0.

Linear Independence

If is not an integer, and are linearly independent. We can see this by focusing on the functions in the vicinity of x = 0. Near x = 0, the negative exponent in indicates that this function is unbounded, while is clearly bounded. Therefore, the two functions are not proportional - thus they must be linearly independent

The left hand side of this relationship is simply the Wronskian of and (with a negative sign), or

Now, for = n where n is an integer, the right hand side of eqn. (8.47) is clearly zero (i.e. for integer n). Therefore, W = 0, and J_n and J_-n are linearly dependent. In fact, it is easy to show from the infinite series representations that, for n an integer,

For not an integer, and , and the two solutions, and , are linearly independent (as shown above). Therefore, when is not an integer the general solution to the ordinary Bessel’s equation becomes

When = n is an integer, we need to develop a second linearly independent solution via some other means.

Ordinary Bessel Functions of the Second Kind

In searching for a second linearly independent solution, consider the following development. For , are linearly independent and eqn. (8.50) represents the general solution to eqn. (8.39). Now let’s define a new function, , in terms of these two linearly independent functions, or

Now since are linearly independent for non-integer , we can write the general solution to Bessel’s equation as

where it is easy to see the correspondence with eqn. (8.50) with values of c₀ and c₁ given by

Now our real interest with these manipulations is to determine what happens when becomes an integer. For this situation, let’s take the limit of eqn. (8.51) as . Performing this operation gives

which, via eqn. (8.49), gives an indeterminate form upon substitution, or

Therefore, to determine this limit, we use L’Hospital’s Rule, or

where it is important to note that the derivative is taken with respect to . Upon actually taking the limit, we have

Now taking the indicated derivatives, item by item, and simplifying, one gets (after considerable manipulation!!!)

with

where is known as Euler’s constant. Although this function is very ugly and extremely tedious to work with in this form, it is, nevertheless, as important function. It is well known and it can be manipulated, evaluated numerically, plotted, differentiated, integrated, etc., just like any other function. and are known as ordinary Bessel functions of the second kind.

Summary Expressions -- Ordinary Bessel Functions

Summary Expressions -- Modified Bessel Functions

In the above table, is referred to as a modified Bessel function of the first kind and is known as a modified Bessel function of the second kind. are linearly independent for any.

Summary Expressions -- Hankel Functions

The Hankel functions of the first and second kind are complex conjugates and they are written as

Additional Properties and Relationships Among the Bessel Functions

One can use the derivative formulas to derive various integral relations. For example, the above expression for , for , gives

Thus, from this relationship, we have

Similarly, the expression for , for , gives

The left hand side of the last expression can be written as the derivative of the product, or

Therefore, integrating this expression gives

Some Plots and Limiting Values for the Low-Order Bessel Functions

Also of interest here are the limiting values of the low-order integer Bessel functions on the interval . In particular, the limiting values can be summarized as follows:

These quantities are particularly useful in evaluating boundary conditions for BVPs which can be solved in terms of integer-order Bessel functions.

The are many more useful relationships for the Bessel functions that have not been tabulated here, and the student is encouraged to browse the literature for a more comprehensive treatise on this subject. We will return to the subject of orthogonality for the ordinary Bessel functions in a later section, and the next subsection gives a recipe for treating a variety of general variable coefficient second-order equations with Bessel function solutions. Beyond this, if the need arises, the reader can always find additional information on this important subject from a variety of sources (there is a lot out there on a wide variety of subjects involving Bessel functions).

Fig. 8.1 Some plots for the low-order Bessel functions.