Two formulae with nodes related to zeros of Bessel functions for semi-infinite integrals: extending Gauss–Jacobi-type rules

Abstract

For approximating integrals $\varvec{\int _0^{\infty }\!\!} \varvec{x}^{\varvec{\alpha }}\varvec{f(x)dx}$ ($\varvec{\alpha >-1}$) over a semi-infinite interval $\varvec{[0,\infty )}$ with a given function $\varvec{f(x)}$, two formulae, one of them new and another associated with an existing formula, are presented. They are constructed in a limiting process to a semi-infinite interval $\varvec{[0,\infty ]}$ with a linear transformation for a well-known approximation method, the Gauss–Jacobi (GJ) rule and its family rules: the Gauss–Jacobi–Radau (GR) and Gauss–Jacobi–Lobatto (GL) rules on a finite interval. This procedure was used in constructing our previous limit Clenshaw–Curtis-type formulae. The limit GJ formula (LGJ) constructed in this way uses as nodes zeros of the Bessel function $\varvec{\varvec{J_{\alpha }(x)}}$ squared after multiplied by a positive constant a and the limit GR (LGR) (and limit GL (LGL)) formula those with zeros of $\varvec{J_{\alpha +1}(x)}$. The LGJ formula is also shown to be derived from the formula developed by Frappier and Olivier (Math. Comp. 60:303–316, 1993) for an integral on $\varvec{[0,\infty )}$. The LGR and LGL formulae give the same and new formula. We show that for a function f(z) analytic on a domain in the complex plane z and satisfying some appropriate conditions, there exists a constant ${d>0}$ such that the errors of both formulae are $\varvec{O(e^{-2d/a})}$ as $\varvec{a\rightarrow +0}$. The average of the LGJ and LGR formulae gives smaller quadrature errors than each formula. Numerical examples confirm these behaviors and show that the LGJ and LGR formulae give asymptotically the same quadrature errors of opposite sign. Consequently, the LGR formula behaves like an anti-LGJ formula in the same way as the Lobatto rule for integrals on $\varvec{[-1,1]}$ behaving like the anti-Gauss rule.

Appendices

Appendix A: Frappier–Olivier formula and LGJ formula

This section shows that the LGJ formula is also derived from a formula (see (8.1) below, slightly modified with real $\alpha >-1$) proposed by Frappier and Olivier [4] (cf. [16,17,18]).

Theorem 8.1

(Frappier and Olivier) Let $B_{\sigma }$ be the class of entire functions of exponential type $\sigma$, bounded on the real axis. For all $f\in B_{2\tau }$ such that $f(x)=O(|x|^{-\delta })$, $x\rightarrow \pm \infty$, with $\delta >2\alpha +2$, there holds

$$\begin{aligned} \int _0^{\infty }\!\!x^{2\alpha +1}(f(x)+f(-x))\,dx= \frac{2}{\tau ^{2\alpha +2}}\sum _{l=1}^{\infty }\frac{j_{\alpha ,l}^{2\alpha }}{[J_{\alpha }{}'(j_{\alpha ,l})]^2} \Big [f\Big (\frac{j_{\alpha ,l}}{\tau }\Big )+ f\Big (-\frac{j_{\alpha ,l}}{\tau }\Big )\Big ], \end{aligned}$$

(8.1)

where $j_{\alpha ,l}$ ($l=\pm 1,\pm 2,\dots$) are the zeros of the entire function $J_{\alpha }(z)/z^{\alpha }$, ordered such that $j_{\alpha ,-l}=j_{\alpha ,l}$ and $j_{\alpha ,1}<j_{\alpha ,2}<\dots$.

Ogata [19] gives a formula for the integrals $\int _{-\infty }^{\infty }|x|^{2\nu +1}\,f(x)\,dx$. Splitting the range $(-\infty ,\infty )$ into $(-\infty ,0]$ and $[0,\infty )$ and setting $t=-x$ in the first integral give the same formula as the right-hand side of (8.1) (the equality is replaced by the approximation $\cong$).

We show that the LGJ formula (1.2) is obtained from (8.1). Applying the change of variables $x=t^2$ to the integral If in (1.1) gives

$$\begin{aligned} \int _0^{\infty }\!\!x^{\alpha }f(x)\,dx=\int _0^{\infty }\!\!2t^{2\alpha +1}f(t^2)\,dt =\int _0^{\infty }\!\!t^{2\alpha +1}[f(t^2)+f((-t)^2)]\,dt. \end{aligned}$$

(8.2)

Using the relation (8.1) in the rightmost-hand side of (8.2), we have

$$\begin{aligned} If=\int _0^{\infty }\!\!x^{\alpha }f(x)\,dx\cong \frac{4}{\tau ^{2\alpha +2}}\sum _{l=1}^{\infty }\frac{j_{\alpha ,l}^{2\alpha }}{[J_{\alpha }{}'(j_{\alpha ,l})]^2} f\Big (\frac{j_{\alpha ,l}^2}{\tau ^2}\Big ). \end{aligned}$$

(8.3)

Setting $\tau =\pi /h$ in the rightmost-hand side of (8.3) gives (1.2) with (1.3).

Appendix B: Proof of Proposition 1.2

In this section, Proposition 1.2 is proved. We claim that for $\alpha =-1/2$, and for nodes $x_{h,l}^{{\textrm{LGJ}}}$ and weights $w_{h,l}^{{\textrm{LGJ}}}$ of the LGJ formula and $x_{h,l}^{{\textrm{LGR}}}$ and $w_{h,l}^{{\textrm{LGR}}}$ of the LGR formula, we have

$$\begin{aligned} x_{h,l}^{{\textrm{LGJ}}}&=\Big [h\Big (l-\frac{1}{2}\Big )\Big ]^2,\quad w_{h,l}^{{\textrm{LGJ}}}=2h\quad (l\ge 1),\end{aligned}$$

(9.1)

$$\begin{aligned} x_{h,0}^{{\textrm{LGR}}}&=0,\quad w_{h,0}^{{\textrm{LGR}}}=h,\quad x_{h,l}^{{\textrm{LGR}}}=(hl)^2,\quad w_{h,l}^{{\textrm{LGR}}}=2h\quad (l\ge 1). \end{aligned}$$

(9.2)

To verify (9.1) and (9.2), we note that the Bessel function $J_{-1/2}(x)$ and its zeros $j_{-1/2,l}$ are given, respectively, by

$$\begin{aligned} J_{-1/2}(x)=\Big (\frac{2}{\pi x}\Big )^{1/2}\cos x,\quad j_{-1/2,l}=(l-1/2)\pi \quad (l\ge 1), \end{aligned}$$

(9.3)

and $J_{1/2}(x)$ and $j_{1/2,l}$ by

$$\begin{aligned} J_{1/2}(x)=\Big (\frac{2}{\pi x}\Big )^{1/2}\sin x, \qquad j_{1/2,l}=l\pi \quad (l\ge 0), \end{aligned}$$

(9.4)

(cf. [4, 5, pp. 54–55]). In view of (9.3) and (9.4), it follows that

$$\begin{aligned} J_{-1/2}'(j_{-1/2,l})=(-1)^l\Big [\frac{2}{\pi ^2(l-1/2)}\Big ]^{1/2}\!\!,\ \ J_{1/2}'(j_{1/2,l})=(-1)^l\Big (\frac{2}{\pi ^2l}\Big )^{1/2}\!\!. \end{aligned}$$

(9.5)

We verify (9.1) by using (9.3) and the first relation of (9.5) in (1.3). We verify (9.2) by using (9.4) and the second relation of (9.5) in (1.5) and (1.6) and since $\Gamma (1/2)\Gamma (3/2)=\pi /2$.

In view of (1.1), (9.1) and (9.2), it follows that

$$\begin{aligned} \int _0^{\infty }\!\!x^{-1/2}f(x)\,dx&\cong 2h\sum _{k=1}^{\infty }f\Big (\Big (h\Big (l-\frac{1}{2}\Big )\Big )^2\Big ),\end{aligned}$$

(9.6)

$$\begin{aligned} \int _0^{\infty }\!\!x^{-1/2}f(x)\,dx&\cong 2h\sum _{l=0}^{\infty }\!'f((hl)^2). \end{aligned}$$

(9.7)

Proposition 1.2 is established by the transformation $x=t^2$ in the left-hand sides of (9.6) and (9.7).$\square$