[Paper Review] On evaluation of integrals involving Bessel functions

D. Babusci, G. Dattoli|arXiv (Cornell University)|Nov 3, 2011

Algebraic and Geometric Analysis1 references3 citations

TL;DR

This paper introduces a symbolic operator method for evaluating definite integrals involving Bessel functions, exponentials, logarithms, and their products—key in quantum field theory. By treating Bessel functions as formal exponentials via umbral calculus, the method enables closed-form evaluation of complex integrals, including those arising in Feynman diagram amplitudes, with results expressed in terms of gamma and digamma functions.

ABSTRACT

We introduce a symbolic method for the evaluation of definite integrals containing combinations of various functions, including exponentials, logarithm and products of Bessel functions of different types. The method we develop is naturally suited for the evaluation of integrals associated with specific Feynman diagrams.

Motivation & Objective

• To develop a systematic symbolic approach for evaluating integrals with Bessel functions, exponentials, and logarithms.
• To address the challenge of divergent integrals by subtracting infinities of the same order, yielding finite results.
• To extend the method to products of Bessel functions of different types and to Macdonald and Neumann functions.
• To enable efficient computation of integrals relevant to Feynman diagram amplitudes in quantum field theory.
• To lay the groundwork for automated symbolic implementation using computational algebra systems.

Proposed method

• Utilizes an umbral operator $\hat{c}$ defined by $\hat{c}^\mu \varphi(0) = \varphi(\mu) = \frac{1}{\Gamma(\mu+1)}$ to represent Bessel functions as formal exponentials.
• Applies the identity $J_\nu(x) = \left(\frac{x}{2}\hat{c}\right)^\nu \exp\left\{-\hat{c}\left(\frac{x}{2}\right)^2\right\} \varphi(0)$ to express Bessel functions algebraically.
• Employs the limit procedure $\lim_{\nu \to -1} \int x^\nu dx = \ln x$ to handle logarithmic terms via regularization of divergent integrals.
• Extends the method to Macdonald functions via $K_n(x) = \frac{\pi}{2} \lim_{\nu \to n} \frac{I_{-\nu}(x) - I_\nu(x)}{\sin(\nu\pi)}$, reformulated using the umbral operator.
• Applies the symbolic method to products of Bessel functions by introducing a generalized operator $\hat{d}$ for $J_\mu(ax)J_\nu(bx)$, leading to a series representation.
• Derives closed-form expressions for integrals like $\int_0^\infty x^\alpha J_\mu(ax)J_\nu(bx) dx$ using generalized hypergeometric-type functions.

Experimental results

Research questions

• RQ1 How can divergent integrals involving $x^\nu$ be regularized to yield finite results, such as $\ln x$?
• RQ2 Can Bessel functions be treated as elementary functions via symbolic operators to simplify integral evaluation?
• RQ3 What is the closed-form expression for $\int_0^\infty x^\mu J_\lambda(px) \ln(bx) dx$?
• RQ4 How can products of Bessel functions of different types be integrated symbolically?
• RQ5 Can the method be extended to integrals involving Macdonald and Neumann functions in quantum field theory?

Key findings

• The integral $\int_0^\infty x^\mu e^{-a x^2} \ln(bx) dx = \frac{1}{4\sqrt{a^{\mu+1}}} \Gamma\left(\frac{\mu+1}{2}\right) \left[\ln\left(\frac{b^2}{a}\right) + \psi\left(\frac{\mu+1}{2}\right)\right]$ is derived using regularization.
• The expression $A_\lambda(p,b) = \int_0^\infty J_\lambda(px) \ln(bx) dx = \frac{1}{p} \left[\ln\left(\frac{2b}{p}\right) + \psi\left(\frac{\lambda+1}{2}\right)\right]$ is obtained via the symbolic method.
• The product $J_\mu(ax)J_\nu(bx)$ is represented as $\left(\frac{a x}{2}\right)^\mu \left(\frac{b x}{2}\right)^\nu \exp\left\{-\hat{d} \left(\frac{x}{2}\right)^2\right\} \varphi_{\mu,\nu}(0;a,b)$, enabling systematic integration.
• The integral $\Omega_{\alpha,\mu,\nu}(a,b) = \int_0^\infty x^\alpha J_\mu(ax)J_\nu(bx) dx$ is evaluated as $2^\alpha a^\mu b^\nu \Gamma\left(\frac{\alpha+\mu+\nu+1}{2}\right) \varphi_{\mu,\nu}\left(-\frac{\alpha+\mu+\nu+1}{2};a,b\right)$.
• The method successfully evaluates $\Xi_{\alpha,\mu,\nu}(a,b) = \int_0^\infty x^\alpha J_\mu(ax)Y_\nu(bx) dx$ using $\Xi = \frac{1}{\sin(\nu\pi)} \left[ \cos(\nu\pi)\Omega_{\alpha,\mu,\nu} - \Omega_{\alpha,\mu,-\nu} \right]$.
• The Tricomi-Bessel function $C_\nu(x)$ is expressed as $\hat{c}^\nu e^{-\hat{c}x} \varphi(0)$, and its integrals are evaluated as $\int_0^\infty C_\nu(x) dx = \frac{1}{\Gamma(\nu)}$ and $\int_0^\infty x^\mu C_\nu(x^2) dx = \frac{1}{2} \frac{\Gamma\left(\frac{\mu+1}{2}\right)}{\Gamma\left(\nu - \frac{\mu-1}{2}\right)}$.