A new Weber type integral equation related to the Weber-Titchmarsh problem

TL;DR

This paper introduces a new solvability condition and closed-form solution for a Weber type integral equation, expanding understanding of Weber-Orr transforms and related classical problems using Mellin transform techniques.

Contribution

The paper presents a novel solution method for Weber type integral equations, connecting classical Weber-Titchmarsh problems with modern Mellin transform techniques.

Findings

Derived solvability conditions for the Weber type integral equation

Obtained closed-form solutions using Mellin transform properties

Connected classical Weber-Titchmarsh problem with modern integral transform methods

Abstract

We derive solvability conditions and closed-form solution for the Weber type integral equation, related to the familiar Weber-Orr integral transforms and the old Weber-Titchmarsh problem (posed in Proc. Lond. Math. Soc. 22 (2) (1924), pp.15, 16), recently solved by the author. Our method involves properties of the inverse Mellin transform of integrable functions. The Mellin-Parseval equality and some integrals, involving the Gauss hypergeometric function are used.

Equations

$$\big{|}\big{|}f\big{|}\big{|}_{{\mathcal{M}}_{c_{1},c_{2}}^{-1}(L_{c})}={1\over 2\pi}\int_{c}e^{\pi c_{1}|s|}|s^{c_{2}}F(s)ds|.$$

$${\mathcal{M}}_{d_1,d_2}^{-1}(L_c)\subseteq {\mathcal{M}}_{c_1,c_2}^{-1}(L_c)$$

$$F_{\nu }(x,s)={\int }_0^{\infty }{\lambda }^{-s}\left[J_{\nu }(x\lambda )Y_{\nu +1}(a\lambda )-Y_{\nu }(x\lambda )J_{\nu +1}(a\lambda )\right]d\lambda =\frac{2^{-s}{a}^{\nu +1}}{\pi x^{2-s+\nu }}\frac{\cos (\pi \nu )}{\Gamma (s/2)}\Gamma (-\nu -1)\Gamma (1+\nu -s/2)$$

$$\times {}_2{F}_1\left(1-\frac{s}{2},1+\nu -\frac{s}{2};2+\nu ;\frac{a^2}{x^2}\right)-\frac{2^{-s}{x}^{\nu +s}}{\pi a^{1+\nu }}\frac{\Gamma (\nu +1)\Gamma (-s/2)}{\Gamma (1+\nu +s/2)}$$

$$\times {}_2{F}_1\left(-\nu -\frac{s}{2},-\frac{s}{2};-\nu ;\frac{a^2}{x^2}\right)-\frac{2^{-s}{a}^{\nu +1}}{\pi x^{1+\nu -s}}\cos \left(\frac{\pi s}{2}\right)\frac{\Gamma (\nu +1-s/2)\Gamma (1-s/2)}{\Gamma (2+\nu )}$$

$$\left|{}_2{F}_1(a,b;c;x)\right|\le \left|\frac{\Gamma (c)}{\Gamma (b)\Gamma (c-b)}\right|{\int }_0^1{u}^{\mathrm{Re}b-1}(1-u{)}^{\mathrm{Re}(c-b)-1}(1-xu{)}^{-\mathrm{Re}a}du$$

$$\left|{}_2{F}_1\left(1-\frac{s}{2},1+\nu -\frac{s}{2};2+\nu ;\frac{a^2}{x^2}\right)\right|\le C{x}^{2-\mathrm{Re}s}(x^2-a^2{)}^{\mathrm{Re}s/2-1}\left|\frac{\Gamma (2+\nu )}{\Gamma ((2(1+\nu )-s)/2)\Gamma ((2+s)/2)}\right|,$$

$$\left|{}_2{F}_1\left(-\nu -\frac{s}{2},-\frac{s}{2};-\nu ;\frac{a^2}{x^2}\right)\right|\le C{x}^{-2\nu -\mathrm{Re}s}(x^2-a^2{)}^{\nu +\mathrm{Re}s/2}\left|\frac{\Gamma (-\nu )}{\Gamma (-s/2)\Gamma (-\nu +s/2)}\right|,$$

$$\left|{}_2{F}_1\left(1+\nu -\frac{s}{2},1-\frac{s}{2};2+\nu ;\frac{a^2}{x^2}\right)\right|\le C{x}^{2(1+\nu )-\mathrm{Re}s}(x^2-a^2{)}^{\mathrm{Re}s/2-1-\nu }$$

$$\times \left|\frac{\Gamma (2+\nu )}{\Gamma ((2-s)/2)\Gamma ((2(1+\nu )+s)/2)}\right|.$$

$$\underset{N\to \infty }{\lim }{\int }_0^N\phi (\lambda )\left[J_{\nu }(x\lambda )Y_{\nu +1}(a\lambda )-Y_{\nu }(x\lambda )J_{\nu +1}(a\lambda )\right]d\lambda ,$$

$${\int }_0^N\phi (\lambda )\left[J_{\nu }(x\lambda )Y_{\nu +1}(a\lambda )-Y_{\nu }(x\lambda )J_{\nu +1}(a\lambda )\right]d\lambda =\frac{1}{2\pi i}{\int }_{\mu -i\infty }^{\mu +i\infty }\Phi (s)F_{\nu }(x,s)ds$$

$$-\frac{1}{2\pi i}{\int }_{\mu -i\infty }^{\mu +i\infty }\Phi (s){\int }_N^{\infty }{\lambda }^{-s}\left[J_{\nu }(x\lambda )Y_{\nu +1}(a\lambda )-Y_{\nu }(x\lambda )J_{\nu +1}(a\lambda )\right]d\lambda ds,x>a.$$

Full text

Recently, the author gave solvability conditions and closed-form solution for the classical Weber equation [7]

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where $f(x)$ is a given function on $[a,\infty),\ a>0,g(x),x\in\mathbb{R}_{+}$ should be determined and the kernel

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involves Bessel functions of the first and second kind $J_{\nu}(z),Y_{\nu}(z),\ \nu\in\mathbb{C}$ [1], Vol. II . It was solved formally by Titchmarsh in 1924 and posed as an open problem (see [3], p. 15.) to describe a class of complex-valued functions $g(x),\ x\in\mathbb{R}_{+}$, which can be expanded in terms of the following repeated integral

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Expansion (3) is related to the familiar Weber-Orr integral expansions of an arbitrary function $f(x)$ as repeated integrals

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which are different from (3). Our method is based on the use of the Mellin transform [4]. Precisely, the Mellin transform is defined in $L_{\mu,p}(\mathbb{R}_{+}),\ 1<p\leq 2$ by the integral

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being convergent in mean with respect to the norm in $L_{q}(\mu-i\infty,\mu+i\infty),\ q=p/(p-1)$. Moreover, the Parseval equality holds for $f\in L_{\mu,p}(\mathbb{R}_{+}),\ g\in L_{1-\mu,q}(\mathbb{R}_{+})$

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The inverse Mellin transform is given accordingly

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where the integral converges in mean with respect to the norm in $L_{\mu,p}(\mathbb{R}_{+})$

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In particular, letting $\mu=1/p$ we get the usual space $L_{1}(\mathbb{R}_{+})$. We will modify the definition of a special class of functions related to the Mellin transform (6) and its inversion (8), which was introduced in [5], [6]. Indeed, we have

Definition 1. Denote by ${\mathcal{M}}^{-1}(L_{c})$ the space of functions $f(x),\ x\in\mathbb{R}_{+}$, being representable by inverse Mellin transform (8) of integrable functions $F(s)\in L_{1}(c)$ on the vertical line $c=\{s\in\mathbb{C}:\mu={\rm Res}=c_{0}\}$.

The space ${\mathcal{M}}^{-1}(L_{c})$ with the usual operations of addition and multiplication by scalar is a linear vector space. If the norm in ${\mathcal{M}}^{-1}(L_{c})$ is introduced by the formula

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then it becomes a Banach space.

**Definition 2 **. Let $\mu\neq 0,\ c_{1},c_{2}\in\mathbb{R}$ be such that $2\hbox{sign}\ c_{1}+\hbox{sign}\ c_{2}\geq 0$. By ${\mathcal{M}}_{c_{1},c_{2}}^{-1}(L_{c})$ we denote the space of functions $f(x),x\in\mathbb{R}_{+}$, representable in the form (8), where $s^{c_{2}}e^{\pi c_{1}|s|}F(s)\in L_{1}(c)$.

It is a Banach space with the norm

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In particular, letting $c_{1}=c_{2}=0$ we get the space ${\mathcal{M}}^{-1}(L_{c})$. Moreover, it is easily seen the inclusion

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when $2\hbox{sign}(d_{1}-c_{1})+\hbox{sign}(d_{2}-c_{2})\geq 0$.

Using this technique we proved the following

Theorem 1  [7]. Let $a>0,\ \nu\in\mathbb{C},\ 0<{\rm Re}\nu<1/2,g(x)\in{\mathcal{M}}_{0,1}^{-1}(L_{c})$ with $c=\{s\in\mathbb{C}:-1<{\rm Re}s<0\}$. Then for almost all $x>0$ expansion $(3)$ holds, where the inner and outer integrals are understood in the improper sense.

These results will be applied to solve the so-called Weber type integral equation

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in the class ${\mathcal{M}}_{0,1}^{-1}(L_{c})$. The key ingredient will be also properties for derivative of Bessel functions, namely, (see [1], Vol. II)

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and the following integral, which is a direct consequence of relation (2.13.15.4) in [2], Vol. 2, namely,

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where $\Gamma(z)$ is Euler’s gamma-function and ${}_{2}F_{1}(a,b;c;x)$ is Gauss’s hypergeometric function [1], Vol. 1, having an integral representation as the Euler integral

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for instance, under conditions ${\rm Re}a>0,\ {\rm Re}c>{\rm Re}b>0\ x\in[0,1).$ Moreover, representation (15) gives us the following uniform estimate for the Gauss function, which will be used below

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where $B(a,b)$ is Euler’s beta-function [1], Vol. 1. Let $-1<{\rm Re}\nu<-1/2$. Then, denoting by the same letter $C$ various positive constants, which can occur, we obtain

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These estimates allow to prove the convergence of the integral (11) as an improper one. In fact, writing it as

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we take $\varphi$ from the subspace ${\mathcal{M}}_{1/2,1}^{-1}(L_{c})\subset{\mathcal{M}}_{0,1}^{-1}(L_{c})$ with $c=\{s\in\mathbb{C}:-1<{\rm Re}s<0\}$. So, according to Definition 2 $\varphi$ is given by integral (8) of some function $\Phi(s)$ from the weighted $L_{1}$-space. Then changing the order of integration by Fubini’s theorem for each fixed $N$ and using (14), it becomes

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We will prove that

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To do this, we appeal to the asymptotic behavior of Bessel functions at infinity [1], Vol. II to find for fixed $x>a$

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Substituting this expression into (17) and integrating by parts in the inner integral with respect to $\lambda$ it gives

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and

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under assumption $\mu+1>0.$ Hence we proved the equality

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where the integral in the right-hand side of (18) converges absolutely. Indeed, from (14), (16) and Stirling’s asymptotic formula for the gamma-function at infinity [1], Vol. I, we have

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Therefore,

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Moreover, it tends to zero when $x\to\infty$ when $\mu-{\rm Re}\nu<0$.

Let $f\in{\mathcal{M}}_{0,1}^{-1}(L_{c})$ with $c=\{s\in\mathbb{C}:{\rm Re}s=\gamma>1/2\}$. Returning to integral equation (11) and observing that due to the absolute and uniform convergence of the integral (8) and its derivative with respect to $x\geq x_{0}>0$ functions $f,\varphi$ are continuously differentiable on $[a,\infty)$ and $\mathbb{R}_{+}$, respectively, we act with the differential operator $\left[x^{\nu}{d\over dx}x^{-\nu}\right]$ on its both sides. Hence, employing (12), (13), we obtain

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It is allowed owing to the uniform convergence by $x\geq a_{0}>a$ of the integral with respect to $\lambda$ in (19) if we keep function $\lambda\varphi(\lambda)$ in the same space ${\mathcal{M}}_{0,1}^{-1}(L_{c})$, i.e.

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where $|s|\Psi(s)\in L_{1}(c),\ c=\{s\in\mathbb{C}:-1<\mu<0\}$. In fact, recalling the asymptotic behavior of Bessel functions at infinity and integrating by parts, we find

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where the differentiation under the integral sign is allowed via the absolute and uniform convergence.

Meanwhile, by the same reasons for $x\geq a$, we have (see (8))

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and it tends to zero when $x\to\infty$ via the estimate

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as well as

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This means that equations (11), (19) are equivalent. Hence employing Theorem 1, the unique solution of equation (19) has the form

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It can be written in a different form with the integration by parts, eliminating outer integrated terms since $t^{1/2}f(t)=o(1),\ t\to\infty$ and taking into account (12), (13). Hence,

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We summarize our results by the following

Theorem 2. Let $-1<{\rm Re}\nu<-1/2$,  $f\in{\mathcal{M}}_{0,1}^{-1}(L_{c})$ with $c=\{s\in\mathbb{C}:{\rm Re}s=\gamma>1/2\},\ \varphi\in{\mathcal{M}}_{1/2,1}^{-1}(L_{c})$ and $\lambda\varphi(\lambda)\in{\mathcal{M}}_{0,1}^{-1}(L_{c})$ with $c=\{s\in\mathbb{C}:-1<{\rm Re}s<{\rm Re}\nu\}$. Then $\varphi$ is the unique solution of the Weber type integral equation $(11)$, given by the formula

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Acknowledgments

The work was partially supported by CMUP [UID/MAT/00144/2013], which is funded by FCT(Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020. The author thanks Mark Craddock for pointing out the Weber type equation for possible applications to his attention.