A Septuple Integral of the Product of Three Bessel Functions of the First Kind J_α(tβ)J_γ(xδ)J_η(yθ): Derivation and Evaluation over General Indices

Abstract

Closed expressions for a number of septuple integrals involving the product of three Bessel functions of the first kind $J_{\alpha }\left(t\beta \right)J_{\gamma }\left(x\delta \right)J_{\eta }\left(y\theta \right)$ when the orders $\alpha ,\gamma ,\eta$ are large, are derived in terms of the Hurwitz–Lerch zeta function $\mathsf{\Phi }(z,s,v)$. The integrals are not easy to to evaluate for complex values of the parameters. All the results in this work are new.

1. Significance Statement

Integrals over Bessel functions [1,2,3,4] have been used in mathematical physics for more than a century. They appear naturally as solutions to the wave equation with cylindrical or spherical symmetry, with the ordinary and spherical Bessel functions involved, respectively. While there is a lot of research on integrals including the product of two Bessel functions, there isn’t much on multiple integrals containing the product of three or more Bessel functions. In this current paper, we will expand upon previous integral formula by deriving a septuple definite integral in terms of the Hurwitz–Lerch zeta function. The parameters will be complex subject to their restrictions and the indices of the Bessel functions are independent of each other. Numerically evaluating such an integral was not easy as the Bessel function is highly oscillatory. We, however, were able to write down a closed form expression which enables the evaluation of the septuple integral involving the product of three Bessel functions.

Given the importance of the integrals whose kernels have the product of three Bessel functions $J_{\alpha }\left(t\beta \right)J_{\gamma }\left(x\delta \right)J_{\eta }\left(y\theta \right)$ encourages us to expand upon this type of work. In this current paper, we derive a septuple integral where the kernel is the product of three Bessel functions, generalized logarithmic and exponential functions. This integral will be derived and represented in terms of the Hurwitz–Lerch zeta function which features analytic continuation for extended evaluation of the parameters involved in the integral transform. Previously published work did not feature a complete analytic solution for the definite integral of the product of these functions, which enhances the importance of the results in the current paper.

2. Introduction

In this paper, we derive the septuple definite integral given by

$$\begin{array}{c}{\int }_{{\mathbb{R}}_{+}^7}{z}^{-m}{q}^{2\alpha -m+1}{r}^{2\eta -m}{t}^{-\alpha +m-1}{v}^{2\gamma +m-1}{x}^{-\gamma -m+1}{y}^{m-\eta }{J}_{\alpha }\left(t\beta \right)J_{\gamma }\left(x\delta \right)J_{\eta }\left(y\theta \right)\hfill \\ \hfill e^{-b\left(q^2+r^2+v^2+z^2\right)}{log}^k\left(\frac{atvy}{qrxz}\right)dxdydzdtdvdrdq\end{array}$$

(1)

where the parameters $k,a,b,\alpha ,\gamma ,\eta ,m$ are general complex numbers and $1/2<Re\left(m\right)<1$, $0\le Im\left(b\right)\le 2\pi ,Re\left(\beta \right)>0,Re\left(\delta \right)>0,Re\left(\theta \right)>0$. This definite integral will be used to derive special cases in terms of special functions and fundamental constants. The derivations follow the method used by us in [5]. This method involves using a form of the generalized Cauchy’s integral formula given by

$$\frac{y^k}{\mathsf{\Gamma }(k+1)}=\frac{1}{2\pi i}{\int }_C\frac{e^{wy}}{w^{k+1}}dw.$$

(2)

where C is in general an open contour in the complex plane where the bilinear concomitant has the same value at the end points of the contour. See [6] for the method.

3. Definite Integral of the Contour Integral

We use the method in [5,6]. Using a generalization of Cauchy’s integral formula, we form the quintuple integral by replacing y by

$$log\left(\frac{atvy}{qrxz}\right)$$

and multiplying by

$$z^{-m}{q}^{2\alpha -m+1}{r}^{2\eta -m}{t}^{-\alpha +m-1}{v}^{2\gamma +m-1}{x}^{-\gamma -m+1}{y}^{m-\eta }{J}_{\alpha }\left(t\beta \right)J_{\gamma }\left(x\delta \right)J_{\eta }\left(y\theta \right)e^{-b\left(q^2+r^2+v^2+z^2\right)}$$

then taking the definite integral with respect to $x\in [0,\infty )$, $y\in [0,\infty )$, $z\in [0,\infty )$, $t\in [0,\infty )$, $u\in [0,\infty )$, $v\in [0,\infty )$ and $q\in [0,\infty )$ to obtain

$$\begin{array}{c}\frac{1}{\mathsf{\Gamma }(k+1)}{\int }_{{\mathbb{R}}_{+}^7}{z}^{-m}{q}^{2\alpha -m+1}{r}^{2\eta -m}{t}^{-\alpha +m-1}{v}^{2\gamma +m-1}{x}^{-\gamma -m+1}{y}^{m-\eta }\hfill \\ J_{\alpha }\left(t\beta \right)J_{\gamma }\left(x\delta \right)J_{\eta }\left(y\theta \right)e^{-b\left(q^2+r^2+v^2+z^2\right)}{log}^k\left(\frac{atvy}{qrxz}\right)dxdydzdtdvdrdq\\ \\ =\frac{1}{2\pi i}{\int }_{{\mathbb{R}}_{+}^7}{\int }_C{a}^w{w}^{-k-1}{z}^{-m-w}{q}^{2\alpha -m-w+1}{r}^{2\eta -m-w}{t}^{-\alpha +m+w-1}{v}^{2\gamma +m+w-1}{x}^{-\gamma -m-w+1}{y}^{-\eta +m+w}\\ J_{\alpha }\left(t\beta \right)J_{\gamma }\left(x\delta \right)J_{\eta }\left(y\theta \right)e^{-b\left(q^2+r^2+v^2+z^2\right)}dwdxdydzdtdvdrdq\\ \\ =\frac{1}{2\pi i}{\int }_C{\int }_{{\mathbb{R}}_{+}^7}{a}^w{w}^{-k-1}{z}^{-m-w}{q}^{2\alpha -m-w+1}{r}^{2\eta -m-w}{t}^{-\alpha +m+w-1}{v}^{2\gamma +m+w-1}{x}^{-\gamma -m-w+1}{y}^{-\eta +m+w}\\ J_{\alpha }\left(t\beta \right)J_{\gamma }\left(x\delta \right)J_{\eta }\left(y\theta \right)e^{-b\left(q^2+r^2+v^2+z^2\right)}dxdydzdtdvdrdqdw\\ \\ =\frac{1}{2\pi i}{\int }_C{\pi }^2{a}^w{w}^{-k-1}csc\left(\pi (m+w)\right){\beta }^{\alpha -m-w}{\delta }^{\gamma +m+w-2}{\theta }^{\eta -m-w-1}{2}^{-\alpha -\gamma -\eta +m+w-3}\\ \hfill b^{-\alpha -\gamma -\eta +m+w-2}dw\end{array}$$

(3)

from Equation (13.24) in [7] and Equation (3.326.2) in [8] where $0<Re(m+w)<Re\left(\alpha \right)+3/2,1/2<Re\left(m\right)<1$ and using the reflection Formula (8.334.3) in [8] for the Gamma function. We are able to switch the order of integration over x, y, z, t and u using Fubini’s theorem for multiple integrals see (9.112) in [9], since the integrand is of bounded measure over the space $\mathbb{C}\times [0,\infty )\times [0,\infty )\times [0,\infty )\times [0,\infty )\times [0,\infty )\times [0,\infty )\times [0,\infty )$.

4. The Hurwitz-Lerch Zeta Function and Infinite Sum of the Contour Integral

In this section, we use Equation (2) to derive the contour integral representations for the Hurwitz–Lerch Zeta function.

4.1. The Hurwitz-Lerch Zeta Function

The Hurwitz–Lerch Zeta function is given by (25.14) in [6,10].

4.2. Infinite Sum of the Contour Integral

Using Equation (2) and replacing y by

$$log\left(a\right)+log\left(b\right)-log\left(\beta \right)+log\left(\delta \right)-log\left(\theta \right)+i\pi (2y+1)+log\left(2\right)$$

then multiplying both sides by

$$-i{\pi }^2{e}^{i\pi m(2y+1)}{\beta }^{\alpha -m}{\delta }^{\gamma +m-2}{\theta }^{\eta -m-1}{2}^{-\alpha -\gamma -\eta +m-2}{b}^{-\alpha -\gamma -\eta +m-2}$$

taking the infinite sum over $y\in [0,\infty )$ and simplifying in terms of the Hurwitz–Lerch Zeta function, we obtain

$$\begin{array}{c}-\frac{i{\pi }^{k+2}{e}^{\frac{1}{2}i\pi (k+2m)}{\beta }^{\alpha -m}{\delta }^{\gamma +m-2}{\theta }^{\eta -m-1}{b}^{-\alpha -\gamma -\eta +m-2}{2}^{-\alpha -\gamma -\eta +k+m-2}}{\mathsf{\Gamma }(k+1)}\hfill \\ \mathsf{\Phi }\left(e^{2im\pi },-k,\frac{-ilog\left(2a\right)-ilog\left(b\right)+ilog\left(\beta \right)-ilog\left(\delta \right)+ilog\left(\theta \right)+\pi }{2\pi }\right)\\ \\ =-\frac{1}{2\pi i}\sum _{y=0}^{\infty }{\int }_{C}i{\pi }^2{a}^w{w}^{-k-1}{e}^{i\pi (2y+1)(m+w)}{\beta }^{\alpha -m-w}{\delta }^{\gamma +m+w-2}{\theta }^{\eta -m-w-1}{2}^{-\alpha -\gamma -\eta +m+w-2}\\ b^{-\alpha -\gamma -\eta +m+w-2}dw\\ \\ =-\frac{1}{2\pi i}{\int }_{C}i{\pi }^2{a}^w{w}^{-k-1}{\beta }^{\alpha -m-w}{\delta }^{\gamma +m+w-2}{\theta }^{\eta -m-w-1}{2}^{-\alpha -\gamma -\eta +m+w-2}{b}^{-\alpha -\gamma -\eta +m+w-2}\\ ·\left[\sum _{y=0}^{\infty }{e}^{i\pi (2y+1)(m+w)}\right]dw\\ \\ =\frac{1}{2\pi i}{\int }_C{\pi }^2{a}^w{w}^{-k-1}csc\left(\pi (m+w)\right){\beta }^{\alpha -m-w}{\delta }^{\gamma +m+w-2}{\theta }^{\eta -m-w-1}{2}^{-\alpha -\gamma -\eta +m+w-3}\\ \hfill b^{-\alpha -\gamma -\eta +m+w-2}dw\end{array}$$

(4)

from Equation (1.232.3) in [8] where $Im\left(\pi \right(m+w\left)\right)>0$ in order for the sum to converge.

5. Definite Integral in Terms of the Hurwitz–Lerch Zeta Function

In this section, we will evaluate Equation (5) for large values of the parameters $\alpha ,\gamma ,\eta$. For computational methods of Bessel functions for large order, see chapter 8 in [7].

Theorem 1.

For all $k,a,b,\alpha ,\gamma ,\eta ,m\in \mathbb{C},1/2<Re\left(m\right)<1,Re\left(\alpha \right)>1,Re\left(\gamma \right)>1,Re\left(\eta \right)>1,$$Re\left(\beta \right)>0,Re\left(\delta \right)>0,Re\left(\theta \right)>0$

$$\begin{array}{c}{\int }_{{\mathbb{R}}_{+}^7}{z}^{-m}{q}^{2\alpha -m+1}{r}^{2\eta -m}{t}^{-\alpha +m-1}{v}^{2\gamma +m-1}{x}^{-\gamma -m+1}{y}^{m-\eta }\hfill \\ J_{\alpha }\left(t\beta \right)J_{\gamma }\left(x\delta \right)J_{\eta }\left(y\theta \right)e^{-b\left(q^2+r^2+v^2+z^2\right)}{log}^k\left(\frac{atvy}{qrxz}\right)dxdydzdtdvdrdq\\ \\ =-i{\pi }^{k+2}{e}^{\frac{1}{2}i\pi (k+2m)}{\beta }^{\alpha -m}{\delta }^{\gamma +m-2}{\theta }^{\eta -m-1}{b}^{-\alpha -\gamma -\eta +m-2}{2}^{-\alpha -\gamma -\eta +k+m-2}\\ \hfill \mathsf{\Phi }\left(e^{2im\pi },-k,\frac{-ilog\left(2a\right)-ilog\left(b\right)+ilog\left(\beta \right)-ilog\left(\delta \right)+ilog\left(\theta \right)+\pi }{2\pi }\right)\end{array}$$

(5)

Proof. 

The right-hand sides of relations (3) and (4) are identical; hence, the left-hand sides of the same are identical too. Simplifying with the Gamma function yields the desired conclusion. □

Example 1.

The degenerate case.

$$\begin{array}{c}{\int }_{{\mathbb{R}}_{+}^7}{z}^{-m}{q}^{2\alpha -m+1}{r}^{2\eta -m}{t}^{-\alpha +m-1}{v}^{2\gamma +m-1}{x}^{-\gamma -m+1}{y}^{m-\eta }\hfill \\ J_{\alpha }\left(t\beta \right)J_{\gamma }\left(x\delta \right)J_{\eta }\left(y\theta \right)e^{-b\left(q^2+r^2+v^2+z^2\right)}dxdydzdtdvdrdq\\ \\ \hfill ={\pi }^{2}csc\left(\pi m\right){\beta }^{\alpha -m}{\delta }^{\gamma +m-2}{\theta }^{\eta -m-1}{2}^{-\alpha -\gamma -\eta +m-3}{b}^{-\alpha -\gamma -\eta +m-2}\end{array}$$

(6)

Proof. 

Use Equation (5) and set $k=0$ and simplify using entry (2) in Table below (64:12:7) in [11]. □

Example 2.

Bierens de Haan integral form.

$$\begin{array}{c}{\int }_{{\mathbb{R}}_{+}^7}\frac{t^{-\alpha -1}{v}^{2\gamma -1}{y}^{-\eta }{z}^{-m-p}{J}_{\alpha }\left(t\right)J_{\gamma }\left(x\right)J_{\eta }\left(y\right)q^{2\alpha -m-p+1}{r}^{2\eta -m-p}{x}^{-\gamma -m-p+1}{e}^{\frac{1}{2}\left(-q^2-r^2-v^2-z^2\right)}}{log\left(\frac{tvy}{qrxz}\right)}\hfill \\ \left(t^m{v}^m{y}^m{q}^p{r}^p{x}^p{z}^p-q^m{r}^m{x}^m{z}^m{t}^p{v}^p{y}^p\right)dxdydzdtdvdrdq\\ \\ \hfill =\pi \left({tanh}^{-1}\left(e^{i\pi p}\right)-{tanh}^{-1}\left(e^{i\pi m}\right)\right)\end{array}$$

(7)

Proof. 

Use Equation (5) and form a second equation by replacing $m\to p$ and taking their difference and simplify after setting $k=-1,a=1$ using entry (3) in Table below (64:12:7) in [11]. □

Example 3.

$$\begin{array}{c}{\int }_{{\mathbb{R}}_{+}^7}\frac{q^{33/4}{r}^{45/4}{v}^{29/3}{J}_4\left(t\right)J_5\left(x\right)J_6\left(y\right)e^{\frac{1}{2}\left(-q^2-r^2-v^2-z^2\right)}}{t^{13/3}{x}^{19/4}{y}^{16/3}{z}^{3/4}log\left(\frac{tvy}{qrxz}\right)}\hfill \\ \left(\sqrt[12]{t}\sqrt[12]{v}\sqrt[12]{y}-\sqrt[12]{q}\sqrt[12]{r}\sqrt[12]{x}\sqrt[12]{z}\right)dxdydzdtdvdrdq\\ \\ \hfill =-\frac{1}{4}\pi log\left(9-6\sqrt{2}\right)\end{array}$$

(8)

Proof. 

Use Equation (7) and set $\alpha \to 4,\gamma \to 5,\eta \to 6,m\to \frac{3}{4},p\to \frac{2}{3}$ and simplify. □

Example 4.

$$\begin{array}{c}{\int }_{{\mathbb{R}}_{+}^7}\frac{q^{111/5}{r}^{166/5}{v}^{103/4}{J}_{11}\left(t\right)J_{13}\left(x\right)J_{17}\left(y\right)e^{\frac{1}{2}\left(-q^2-r^2-v^2-z^2\right)}}{t^{45/4}{x}^{64/5}{y}^{65/4}{z}^{4/5}log\left(\frac{tvy}{qrxz}\right)}\hfill \\ \left(\sqrt[20]{q}\sqrt[20]{r}\sqrt[20]{x}\sqrt[20]{z}-\sqrt[20]{t}\sqrt[20]{v}\sqrt[20]{y}\right)dxdydzdtdvdrdq\\ \\ \hfill =\frac{1}{2}\pi {tanh}^{-1}\left(\sqrt{2}\left(1+\frac{4}{-8+\sqrt{2}+\sqrt{10}}\right)\right)\end{array}$$

(9)

Proof. 

Use Equation (7) and set $\alpha \to 11,\gamma \to 13,\eta \to 17,m\to \frac{3}{4},p\to \frac{4}{5}$ and simplify. □

Example 5.

$$\begin{array}{c}{\int }_{{\mathbb{R}}_{+}^7}\frac{q^{41/5}{r}^{26/5}{v}^{11/3}{J}_4\left(t\right)J_2\left(x\right)J_3\left(y\right)e^{\frac{1}{2}\left(-q^2-r^2-v^2-z^2\right)}}{t^{13/3}{x}^{9/5}{y}^{7/3}{z}^{4/5}log\left(\frac{tvy}{qrxz}\right)}\hfill \\ \left(t^{2/15}{v}^{2/15}{y}^{2/15}-q^{2/15}{r}^{2/15}{x}^{2/15}{z}^{2/15}\right)dxdydzdtdvdrdq\\ \\ \hfill =\frac{1}{4}\pi log\left(\frac{1}{3}\left(5+2\sqrt{5}\right)\right)\end{array}$$

(10)

Proof. 

Use Equation (7) and set $\alpha \to 4,\gamma \to 2,\eta \to 3,m\to \frac{4}{5},p\to \frac{2}{3}$ and simplify. □

6. Invariant Index Forms

Example 6.

The polylogarithm function $L{i}_n\left(z\right)$.

$$\begin{array}{c}{\int }_{{\mathbb{R}}_{+}^7}{z}^{-m}{q}^{2\alpha -m+1}{r}^{2\eta -m}{t}^{-\alpha +m-1}{v}^{2\gamma +m-1}{x}^{-\gamma -m+1}{y}^{m-\eta }\hfill \\ J_{\alpha }\left(t\right)J_{\gamma }\left(x\right)J_{\eta }\left(y\right)e^{\frac{1}{2}\left(-q^2-r^2-v^2-z^2\right)}{log}^k\left(-\frac{tvy}{qrxz}\right)dxdydzdtdvdrdq\\ \\ \hfill =-i{2}^k{\pi }^{k+2}{e}^{\frac{1}{2}i\pi (k-2m)}{Li}_{-k}\left(e^{2im\pi }\right)\end{array}$$

(11)

Proof. 

Use Equation (5) and set $a\to -1,b\to \frac{1}{2},\beta \to 1,\delta \to 1,\theta \to 1$ and simplify using Equation (25.14.3) in [10]. □

Example 7.

The constant $log\left(2\right)$.

$$\begin{array}{c}{\int }_{{\mathbb{R}}_{+}^7}\frac{q^{2\alpha +\frac{1}{4}}{r}^{2\eta -\frac{3}{4}}{t}^{-\alpha -\frac{1}{4}}{v}^{2\gamma -\frac{1}{4}}{x}^{\frac{1}{4}-\gamma }{y}^{\frac{3}{4}-\eta }{J}_{\alpha }\left(t\right)J_{\gamma }\left(x\right)J_{\eta }\left(y\right)e^{\frac{1}{2}\left(-q^2-r^2-v^2-z^2\right)}}{z^{3/4}log\left(-\frac{tvy}{qrxz}\right)}dxdydzdtdvdrdq\hfill \\ \\ \hfill =-\frac{1}{8}{(-1)}^{3/4}\pi (\pi -2ilog\left(2\right))\end{array}$$

(12)

Proof. 

Use Equation (11) and set $k\to -1,m\to 3/4$ and simplify. □

Example 8.

Catalan’s constant C.

$${\int }_{{\mathbb{R}}_{+}^7}\frac{q^{2\alpha +\frac{1}{4}}{r}^{2\eta -\frac{3}{4}}{t}^{-\alpha -\frac{1}{4}}{v}^{2\gamma -\frac{1}{4}}{x}^{\frac{1}{4}-\gamma }{y}^{\frac{3}{4}-\eta }{J}_{\alpha }\left(t\right)J_{\gamma }\left(x\right)J_{\eta }\left(y\right)e^{\frac{1}{2}\left(-q^2-r^2-v^2-z^2\right)}}{z^{3/4}{log}^2\left(-\frac{tvy}{qrxz}\right)}dxdydzdtdvdrdq=\frac{1}{192}{(-1)}^{3/4}\left({\pi }^2+48iC\right)$$

(13)

Proof. 

Use Equation (11) and set $k\to -2,m\to 3/4$ and simplify using Equation (2.2.1.2.7) in [12]. □

Example 9.

The Riemann zeta function $\zeta \left(s\right)$.

$$\begin{array}{c}{\int }_{{\mathbb{R}}_{+}^7}\frac{q^{2\alpha +\frac{1}{2}}{r}^{2\eta -\frac{1}{2}}{t}^{-\alpha -\frac{1}{2}}{v}^{2\gamma -\frac{1}{2}}{x}^{\frac{1}{2}-\gamma }{y}^{\frac{1}{2}-\eta }{J}_{\alpha }\left(t\right)J_{\gamma }\left(x\right)J_{\eta }\left(y\right)e^{\frac{1}{2}\left(-q^2-r^2-v^2-z^2\right)}}{\sqrt{z}}\hfill \\ {log}^k\left(-\frac{tvy}{qrxz}\right)dxdydzdtdvdrdq\\ \\ \hfill =-2^k\left(2^{k+1}-1\right)e^{\frac{i\pi k}{2}}{\pi }^{k+2}\zeta (-k)\end{array}$$

(14)

Proof. 

Use Equation (11) and set $m\to 1/2$ and simplify using Equations (25), (12) and (10) in [10]. □

Example 10.

Apery’s constant $\zeta \left(3\right)$.

$${\int }_{{\mathbb{R}}_{+}^7}\frac{q^{21/2}{r}^{47/2}{v}^{35/2}{J}_5\left(t\right)J_9\left(x\right)J_{12}\left(y\right)e^{\frac{1}{2}\left(-q^2-r^2-v^2-z^2\right)}}{t^{11/2}{x}^{17/2}{y}^{23/2}\sqrt{z}{log}^3\left(-\frac{tvy}{qrxz}\right)}dxdydzdtdvdrdq=\frac{3i\zeta \left(3\right)}{32\pi }$$

(15)

Proof. 

Use Equation (11) and set $k\to -3,\alpha \to 5,\gamma \to 9,\eta \to 12$ and simplify. □

Example 11.

The constant $\zeta \left(5\right)$.

$${\int }_{{\mathbb{R}}_{+}^7}\frac{q^{17/2}{r}^{43/2}{v}^{27/2}{J}_4\left(t\right)J_7\left(x\right)J_{11}\left(y\right)e^{\frac{1}{2}\left(-q^2-r^2-v^2-z^2\right)}}{t^{9/2}{x}^{13/2}{y}^{21/2}\sqrt{z}{log}^5\left(-\frac{tvy}{qrxz}\right)}dxdydzdtdvdrdq=-\frac{15i\zeta \left(5\right)}{512{\pi }^3}$$

(16)

Proof. 

Use Equation (11) and set $k\to -5,\alpha \to 4,\gamma \to 7,\eta \to 11$ and simplify. □

7. Conclusions

In this paper, we have presented a novel method for deriving a new integral transform containing the product of three Bessel Functions of the First Kind $J_{\alpha }\left(t\beta \right)J_{\gamma }\left(x\delta \right)J_{\eta }\left(y\theta \right)$ along with some interesting definite integrals, using contour integration. The results presented were numerically verified for both real and imaginary, and complex values of the parameters in the integrals using Mathematica by Wolfram.