Integrals of Bessel Functions

CHAPTER I

BASIC FORMULAS

1.1. Introduction

The purpose of this chapter is to collect numerous formulas needed to establish results of subsequent chapters, and to aid in the evaluation of these and other expressions. Some general sources for this material are Erdélyi et al. (1953, 1954), Kratzer and Franz (1960), Rainville (1960), Watson (1945) and Whittaker and Watson (1927). Also, for tables of integrals and related data, see Gröbner and Hofreiter (1949, 1950), Magnus and Oberhettinger (1948, 1954), Meyer zur Capellen (1950) and Ryshik and Gradstein (1957). For further material on Bessel functions and integrals of Bessel functions, see Bowman (1958), Gray, Mathews and MacRobert (1952), McLachlan (1955), Petiau (1955), Relton (1946), Rey Pastor and Castro Brzezicki (1958), Weyrich (1937), Wheelon and Robacker (1954).

Description of numerical mathematical tables of integrals involving Bessel functions is usually given at the end of the chapter where the pertinent material is discussed. In some instances, especially in Chapter XIII, this information is integrated with a discussion of the tabulated functions. We have attempted to give thorough coverage of tables since 1945 approximately. Some, but not all prior data are also referenced. For the sake of completeness, some references for tables of Bessel functions are presented in 1.4.8. As most of these items are standard and well known, usually only an abbreviated description of the contents of each reference is provided. For description of numerical mathematical tables., not only of integrals involving Bessel functions, but also of the entire spectrum of transcendental functions, see Bateman and Archibald (1944), Fletcher, Miller and Rosenhead (1946), Lebedev and Feodorova (1956), Burunova (1959), the journal Mathematical Tables and Other Aids to Computation (now Mathematics of Computation) (1943-present), National Bureau of Standards (1962) and Schütte (1955).

Collections of analytical descriptions and numerical tables of transcendental functions are provided in Jahnke and Emde (1945), Jahnke, Emde and Lösch (1960) and National Bureau of Standards (1962).

1.2. The Gamma Function and Related Functions

For 2F1, see 1.3.

Some useful constants are as follows.

1.3. Generalized Hypergeometric Series

1.3.1. Definition and Basic Properties

We consider the series

Where no confusion will result, we notate the above series as _PFq. It is often convenient to employ a contracted notation and write

Thus is interpreted as and as etc. An empty term is interpreted as unity. For example, if p = 2, (ap)k for p>2.

pFq is not defined if any bq is a negative integer. It terminates if any ap is a negative integer or zero. If pFq does not terminate, it converges for all finite z if p ≤ q, converges for |z| < 1 if p = q+1, converges for z = 1 if p = q+1 and diverges for all z ≠ 0 if p > q+1.

If δ is the operator zD, D = d/dz, then u = pFg is a solution of the differential equation

If p ≤ q+1, then

are the (q+l) linearly independent solutions of (4) provided that no two of the bi’s differ by a negative integer or zero. In the singular cases just mentioned, solutions can be constructed as in the case of Gauss’ series, Bessel functions, etc. We omit further details, but see Erdélyi et al. (1953, Vol. I, Chs. 2,6 and 7) and the references quoted there. See also Chapter VIII where a study is made of a particular 2F3. If p ≥ q+1, there are p formal solutions proportional to Lpq, t = 1,2,...,p, see 1.3.3(8).

1.3.2. Integral Representations

where he poles of to the right of the contour while those of lie to the left of the contour.

1.3.3. Asymptotic Expansions

Consider the divergent series

and denote the sum of the first (n+1) terms by Sn(z). With certain restrictions on arg z, (1) is an asymptotic expansion of f(z) if for n fixed,

That is, we can make with arbitrarily small and |z| sufficiently large. The notation

means that (1) is the asymptotic expansion of f(z). An alternative notation is

We now present asymptotic expansions of pFq for and 0 ≤ p ≤ q. We follow the work of Meijer (1946) which should he consulted for further details. Assume that no ap is a negative integer or zero. Let

and for the determination of the Nk’s, see the discussion surrounding (19)-(38) ahead.

Define

where aj-at. is not an integer or zero for all j ≠ t. This restriction may be removed, but further discussion is deferred to 1.3.4. Let

The asymptotic representations are divided into two cases.

First Case. 0 ≤. p ≤. q-1.

If 0 ≤ p ≤ q-1, then

In the latter two equations, only the dominant expressions are recorded. There are (q - p+l) terms of exponential type, i.e., of the form Kp,q, and the omitted ones are of lower order than those given above. To complete this case, we need the result where p = q-1. In this instance, we give a single expression which includes both the dominant and subdominant parts. Thus

For numerous applications, it is convenient to replace z by and write

Also

The apparent discrepancy in (14) when z has a value such that is a case of Stoke’s phenomenon. See Watson (1945, p.20l).

Second Case, p = q≥l.

Again we give the complete representation.

Also

If , the apparent discrepancy in (17) is again a case of Stoke’s phenomenon.

The following is an important corollary of the above expansions. If p = q≥l or if q-1 = p≥O, then

Evaluation of the coefficients Nk is next of interest. In this connection note that in the Kpq(z) term associated with the asymptotic representation of ui(z), see 1.3.1(5), the coefficients Nk are independent of i. The Nk’s are conveniently found by constructing a formal series solution of 1.3.1(4) using (5) and (10). For later use, we record in complete detail the expansions Kpq(z) for the cases q = p+1 where p = 0,1 and 2 ; and p = q, where p = 1 and 2. These are as follows.

Case I. q = p+1.

From (5) and (6), with z replaced by , we have

Case I.1. p = 0, q = 1.

If b1 = ν,

and this leads to the asymptotic expansions of Bessel functions. See 1.4.6(1-6).

Case I.2. p = 1, q = 2.

If a-L = 1 and b2 – b1 = ν, then ck is given by (21).

Case I.3. p = 2, q = 3.

Case II. p = q.

Replace z by 2z. Then (5) and (6) yield

Case II.1. p = q = 1.

and this also leads to asymptotic expansions for Bessel functions. See the concluding remarks of 1.4.6.

Case II.2. p = q = 2.

If

then ck is given by (21).

1.3.4. The Form of Lp,q(z) for Special Values of the Parameters

The definition of Lp,q(z) by 1.3.3(7-8) conveniently supposes that aj-at is not an integer or zero for all j ≠ t. If this is not so, Lp,q(z) still has meaning and its representation must be found by a limiting process. We now obtain the representation in a special case. Assume first that no pair of aj. values differ by an integer or zero except a1_ and a2. Let where m is a positive integer or zero.

Before stating and proving the main result, it is helpful to introduce further notation. Let

where it is understood that the notation is compact as in 1.3.1(3), and further that short for etc. Then

where m = a2-a1 is a positive integer or zero. Here it is understood that p ≠ 1, p ≠ 2 in the compact notation etc.

In the applications, it happens often that a2 = bq for some particular value of q. In particular, suppose q = 1. Then

Again the notation is compact and in etc., q = 2,3,...q..

To prove (3), we write

Now

Apply this to the first q+1Fp-1 in (6). Then with the aid of the reflection formula for gamma functions, we find

where

Use L’Hospital’s theorem to evaluate A(0). Then this result together with (1), (2) and (5)-(6) readily leads to the assertion (3).

Put (a2-b1) = δ in the third expression of (3). Note that (δ)k = 1 if k = 0 and if k>0. Also if k = 0 and , k>0. Thus the readily simplifies and (4) follows. Alternatively, we can put in (9). Then the q+2Fp is unity and application of L’Hospital’s theorem, etc., gives (4).

If another pair of aj’s differ by a positive integer or zero, say a4-a3=m1 but (a4-a1) is not an integer or zero, then the above limiting analysis can be applied to evaluate and further like extensions for other pairs of aj’s are apparent.

If a2-a1_ and a3-a2 are positive integers or zero, but no ap-aj, j = 1,2,3, p ≠ j is a positive integer or zero, then as before we can determine