Bessel functions of the first kind

Integral representation for Bessel’s functions of the first kind and Neumann series

Abstract.

A Fourier-type integral representation for Bessel’s function of the first kind and complex order is obtained by using the Gegenbuaer extension of Poisson’s integral representation for the Bessel function along with a trigonometric integral representation of Gegenbauer’s polynomials. This representation lets us express various functions related to the incomplete gamma function in series of Bessel’s functions. Neumann series of Bessel functions are also considered and a new closed-form integral representation for this class of series is given. The density function of this representation is simply the analytic function on the unit circle associated with the sequence of coefficients of the Neumann series. Examples of new closed-form integral representations of special functions are also presented.

Key words and phrases:
Bessel function of the first kind, Neumann series of Bessel functions, Integral representations
2010 Mathematics Subject Classification:
33C10,40C10,33B20

1. Introduction

The Bessel function of the first kind and order is defined by the series [27, Eq. 8, p. 40]

(1.1)

which is convergent absolutely and uniformly in any closed domain of and in any bounded domain of . It is the solution of Bessel’s equation

(1.2)

which is nonsingular at . The function is therefore an analytic function of for any , except for the branch point if is not an integer.

In this paper, a new integral representation for the Bessel functions of the first kind and order , , is obtained. This representation, which is given in Section 3, generalizes to complex values of the order the classical Bessel’s integral

(1.3)

which holds only for integral order values (see, e.g., formula (3.21)). The Bessel functions of the first kind are expressed in terms of an integral of Fourier-type, which involves the regularized incomplete gamma function , with being the (complex) fractional part of . This result is achieved by using the Poisson integral representation of (, ) given in terms of Gegenbauer polynomials and then by exploiting the integral representation of the latter polynomials that we prove in Section 2.

The Fourier form of the integral representation of motivates us to consider the Fourier inversion formula in Section 4. This analysis allows us to obtain, for either integer and half-integer orders , a trigonometric expansion of the lower incomplete gamma function, whose coefficients are related to (modified and unmodified) Bessel functions of the first kind. Thus, classical Bessel series expansions of various special functions which are connected with the incomplete gamma function, can be easily obtained in a unified form.

A Neumann series of Bessel functions is an expansion of the type

(1.4)

where are given coefficients. Many special functions of the mathematical physics enjoy an expansion of this type, e.g., Kummer confluent hypergeometric’s, Lommel’s , Kelvin’s, Whittaker’s, and so on. In Section 5 we exploit the novel representation of Bessel’s functions presented in Section 3 in order to derive a simple closed-form integral representation of expansions (1.4). The set of coefficients , which characterizes the series (1.4), comes into this integral representation in a very simple way through the associated analytic function in the unit disk , i.e.: with a weak restriction on the sequence of coefficients , which is sufficient to guarantee the boundedness of on the unit circle. Examples of special functions with this kind of closed-form integral representation are also given.

2. Connection between the coefficients of Gegenbauer and Fourier expansions

The Gegenbauer (ultraspherical) polynomials of order () may be defined by means of the generating function [23, Eq. (4.7.23)] (see also [23, Eq. (4.7.6)]):

(2.1)

For fixed , the Gegenbauer polynomials are orthogonal on the interval with respect to the weight function , that is:

(2.2)

where:

(2.3)

Particularly important special cases of Gegenbauer polynomials are obtained for , which gives the Legendre polynomials , for yielding the Chebyshev polynomials of the second kind , and in a suitable limiting form for the Chebyshev polynomials of the first kind [23]: , where denotes the Pochhammer symbol.

For they admit the following integral representation [26, Eq. (1), p. 559]:

(2.4)

Now, our aim in this section is to present an integral representation of the Gegenbauer polynomials. To this end, we prove the following proposition (see also [6, 9]).

Proposition 2.1.

For , the following integral representation for the Gegenbauer polynomials (, ) holds:

(2.5)
Proof.

In representation (2.4) we first introduce the variable , defined by ():

(2.6)
Figure 1. Integration path for the evaluation of the integral representation (2.5) of the Gegenbauer polynomials.

Then, in the integral in (2.6) we substitute to the complex integration variable defined by

It can be checked that

(2.7)

Now, since , the integrand on the right-hand side of (2.6) can be written as follows:

(2.8)

In order to determine the integration path, consider the intermediate step where is chosen as integration variable; the original path (corresponding to ) is the (oriented) linear segment starting at and ending at . Since (as shown by (2.8)) the integrand is an analytic function of in the disk (since ), the integration path can be replaced by the circular path (see Fig. 1). Moreover, by using the fact that is positive for and therefore at , we conclude from the left equality in (2.7) that in the r.h.s. of (2.8) the following specification holds (for ):

Finally, accounting for this latter expression, the integral representation (2.5) follows directly from (2.6). ∎

Remark 1.

In the proof of Proposition 2.1 we could even choose the integration path , oriented as shown in Fig. 1. Following the same arguments given above, it is easily seen that this option yields the following alternative integral representation of the Gegenbauer polynomials [6, 8]:

(2.9)

3. Fourier-type integral representation of Bessel functions of the first kind

It is well-known that the Bessel functions of the first kind of integral order can be represented by Bessel’s trigonometric integral (1.3) (see also [11, Eq. 10.9.2]). For , satisfies the symmetry relation: . This latter relation no longer holds when is not an integral number since, for , and are linearly independent solutions of the Bessel equation of order . Numerous formulae express the Bessel functions of the first kind as definite integrals, which can be exploited to obtain, for instance, approximations and asymptotic expansions (see [27, Chapter VI], [12], and the website [11, Sect. 10.9] for a useful collection of formulae). Our goal in what follows is to generalize the trigonometric representation (1.3) in order to obtain a Fourier-type integral representation for the Bessel functions of the first kind with holds for complex order with . We can prove the following theorem.

Theorem 3.1.

Let be a non-negative integer, and any complex number such that . Then, the following integral representation for the Bessel functions of the first kind holds:

(3.1)

where the -periodic function is given by

(3.2)

and is the sign function, is the complex fractional part of , denotes the regularized incomplete gamma function, being the lower incomplete gamma function.

Proof.

We start from Gegenbauer’s generalization of Poisson’s integral representation of the Bessel functions of the first kind [27, §3.32, Eq. (1), p. 50]:

(3.3)

where is the Gegenbauer polynomial of order and degree . Formula (3.3) holds for and . Now, we plug (2.5) into (3.3) and, using the Legendre duplication formula for the gamma function, we obtain:

(3.4)

Interchanging the order of integration, (3.4) can be written as follows:

(3.5)

Next, changing the integration variables: and , the second integral on the r.h.s. of (3.5) becomes:

which, inserted in (3.5), yields:

(3.6)

Formula (3.6) can then be written as follows:

(3.7)

where

(3.8)

Then, changing in (3.8) the integration variable and recalling that for , , can be rewritten as follows:

(3.9)

The regularized gamma function enjoys the following recurrence relation [11, Eq. 8.8.11]:

(3.10)

Let us set as the integer part of , i.e.: , where

(3.11)

Moreover, we denote by the complex fractional part of , defined as: , with . We can then insert formula (3.10) into (3.9) and obtain:

(3.12)

When plugged into (3.7) the second term in the squared brackets of (3.12) (which is different from zero only if ) contributes with a (finite) linear combination of the following integrals:

(3.13)

which are null for and . Therefore, finally reads

(3.14)

The -domain of representation (3.7) can be analytically extended to the half-plane where, for any , the integrand is analytic on and the integral defines a function which is locally bounded on every compact subsets of . To see this, it is useful to make explicit the singularity brought by the regularized gamma function by writing the latter in terms of Tricomi’s form of the incomplete gamma function, i.e.: , where

(3.15)

is an entire function in as well as in [24, Eq. (2)]. From (3.7) and (3.14) we thus obtain:

(3.16)

For negative values of , the integrand in (3.16) has a singularity in due to the term , which is integrable for . Hence, the integral (3.16) defines an analytic function of on and representation (3.1) holds true for . ∎

Note 1.

Alternatively, if in the Abel-type integral in (3.8) we change the integration variable the function can be written in terms of Fresnel-type integrals , i.e:

(3.17)

where

(3.18)

and the classical Fresnel integrals and can be obtained as real and imaginary part of (3.18) with , respectively.

Representation (3.1) can be reformulated as follows:

(3.19)

In the last integral we now change the integration variable: , and obtain for :

(3.20)

Now, we can put , with . Since , from (3.20) we therefore obtain the following integral representation of the Bessel function of the first kind:

(3.21)

As mentioned earlier, formula (3.21) (and formulae (3.1) and (3.2) as well) generalizes to complex values of the classical Bessel integral (1.3), which holds for integral values of . In fact, if in (3.21) we put integer, then and, since (see [14, Eq. (2.2)]), formula (1.3) readily follows.

In view of the relation: for [11, Eq. 10.27.6], we have also the following integral representation for the modified Bessel function of the first kind .

Corollary 3.2.

Let be any complex number such that . Then the following integral representation for the modified Bessel function of the first kind holds ():

(3.22)

Formula (3.22) generalizes to complex values of , , the well-known representation [11, Eq. 10.32.3]:

(3.23)

which holds for integral values of .

4. Inversion of the Fourier representation

Representation (3.1) shows that, for fixed , the function coincides with the th Fourier coefficient () of the -periodic function . We are thus prompted to consider the following trigonometrical series:

(4.1)

where denote, for fixed , the th Fourier coefficients of :

(4.2)

However, the Fourier sum representation (4.1) of can actually be written explicitly only in some specific cases for the lack of knowledge of the Fourier coefficients with . Indeed, equation (3.1) states that only for . However, from equation (3.2) we see that (for fixed ) enjoys the following symmetry:

(4.3)

which, substituted in (4.2), gives

(4.4)

Formula (4.4) can thus induce a -index symmetry on the Fourier coefficients only if is integer, i.e.: can be either integer or half-integer ().

Case integer. If is a nonnegative integer, formula (4.4) yields:

(4.5)

For all the Fourier coefficients , , can be determined directly from (3.1) since . For , the Fourier coefficients with cannot be obtained by means of the above symmetry relation, but can nevertheless be expressed as (linear) functionals of the set [6]. Since and , from (3.2) we have: , which inserted in (4.1), and using (4.5), gives:

(4.6)

where for . If we put , formula (4.6) yields:

(4.7)

which represents the well-known Jacobi-Anger expansion of a plane wave into a series of cylindrical waves [11, Eq. 10.12.2].

Case half-integer. If is a positive half-integer, , , the symmetry formula (4.4) reads:

(4.8)

For the Fourier coefficients with can be obtained from those with from the symmetry relation: , . Instead, as in the case of integral , if the Fourier coefficients with are expressible as linear functionals of the coefficients with .

Now, we can plug formula (3.2) for into (4.1) and, in view of (4.8), obtain the following expansion for the incomplete gamma function of order :

(4.9)

Now, putting in (4.9) , and we obtain:

(4.10)

where we used the modified Bessel functions of the first kind . Finally, recalling that , where denotes the error function [11, Eq. 7.2.1], we obtain the following expansion:

(4.11)

A similar representation of the error function in terms of modified Bessel functions of the first kind of integer order is given by Luke in [19, Eq. (2.11)] through the expansion of the confluent hypergeometric function in series of Bessel functions. As a particular case of (4.11), we first put and obtain (see also [20, Eq. (20)]):

(4.12)

Similarly, if in (4.11) we put and , we get the expansion:

(4.13)

which is equivalent to [11, Eq. 7.6.8] (see also [20, Eq. (1), p. 122] and [25, Eqs. (33) and (34)]).

Finally, as a last example of application of expansion (4.1), we can use the form (3.17) of given in terms of Fresnel integrals. We can set in (3.17) and (4.1) and put , , then we obtain the following representation of Fresnel’s integral [11, Eqs. 7.2.7 and 7.2.8] (see also (3.18)):

(4.14)

where denotes the Chebyshev polynomials of the first kind and are the spherical Bessel functions of the first kind: .

5. Neumann series of Bessel functions

Neumann series of Bessel functions are defined by [27, Chapter XVI]

(5.1)

where is in general a complex variable, and and are constants. This kind of series have been investigated extensively in view of their relevance in a number of physical problems (see the Introduction of Ref. [21] for a brief summary of these problems). In addition, Neumann series have been shown to be a useful mathematical tool to the solution of classes of differential and mixed differences equations (see, e.g., [18] and [27, p. 530]). The domain of convergence of series (5.1) is a disk whose radius evidently depends on the asymptotic behaviour of the sequence of coefficients and can be determined by the condition

(5.2)

Integral representations are powerful tools to deduce properties of series. In the case of expansions of type (5.1), examples of integral representations have been given by Wilkins [28], Rice [22] and, more recently, by Pogány, Süli and coworkers [3, 15, 21]. In this section we give an integral representation of the function , which the Neumann series converges to. This is achieved by exploiting the representation (3.1) of given is Section 3, and assuming the condition on the asymptotic behavior of the sequence which basically guarantees the boundedness of trigonometric series. The Fourier-type integral representation of allows us to write in integral form with a kernel (the kernel associated with Neumann expansions) which is proportional to the regularized incomplete gamma function and is independent of the coefficients . The density function of the integral representation, instead, depends only upon the given input sequence of coefficients , and is simply the analytic function, on the unit circle, associated with the sequence of coefficients . This representation easily leads to closed-form integral representations of Neumann expansions of Bessel functions whenever the closed-form of the analytic function associated with the sequence of coefficients is known. As examples of application of this result, new integral representations for bivariate Lommel’s functions and Kelvin’s functions will then be given. More examples of this type of integral representations of special functions of the mathematical physics will be presented and analyzed in a forthcoming paper [10]. Now, we can prove the following theorem.

Theorem 5.1.

Let be a null sequence such that

(5.3)

Then, for the series (5.1) has the following integral representation in the slit domain if or in if :

(5.4)

where

(5.5)

and the kernel is given by

(5.6)