Evaluation of improper integrals by method of hybrid integral transform of Fourier-(Kontorovich-Lebedev)-Fourier type on the polar axis (Q2768786)

This paper deals with solutions on the set \(I_{2}^{+}=\{r: r\in(R_0,R_1)\cup(R_1,R_2)\cup(R_2,\infty); R_0>0\}\) of the system of Fourier and Bessel equations NEWLINE\[CARRIAGE_RETURNNEWLINE\displaystyle\left({d^2\over dr^2}-q_1^2 \right)u_1(r)=-g_1(r),\;r\in(R_0,R_1),\;\left(B_{\alpha}-q_2^2 \right)u_2(r)=-g_2(r),\;r\in(R_1,R_2), CARRIAGE_RETURNNEWLINE\]NEWLINE NEWLINE\[CARRIAGE_RETURNNEWLINE \displaystyle \left({d^2\over dr^2}-q_3^2 \right)u_3(r)=-g_3(r),\;r\in(R_2,\infty),CARRIAGE_RETURNNEWLINE\]NEWLINE with the boundary conditions NEWLINE\[CARRIAGE_RETURNNEWLINE\left.\left(\alpha_{11}^0{d\over dr}+\beta_{11}^0\right)u_1(r)\right|_{r=R_0}=g_0,\;\left.{du_3\over dr}\right|_{r=\infty}=0,CARRIAGE_RETURNNEWLINE\]NEWLINE and the conjunction conditions NEWLINE\[CARRIAGE_RETURNNEWLINE\left.\left[\left(\alpha_{j1}^{k}{d\over dr}+\beta_{j1}^{k}\right)u_{k}(r)-\left(\alpha_{j2}^{k}{d\over dr}+\beta_{j2}^{k}\right)u_{k+1}(r)\right]\right|_{r=R_{k}}=0,\;j,k=1,2.CARRIAGE_RETURNNEWLINE\]NEWLINE Here \(|\alpha_{11}^0|+|\beta_{11}^0|\neq 0,\;\alpha_{jm}^{k}\geq 0, \beta_{jm}^{k}\geq 0, c_{1k}c_{2k}>0, c_{jk}=\alpha_{2j}^{k}\beta_{1j}^{k}-\alpha_{1j}^{k}\beta_{2j}^{k}, j,m,k=1,2,3\); \(2\alpha+1\geq 0, \lambda\in(0,\infty)\); \(B_{\alpha}=r^2d^2/dr^2+(2\alpha+1)rd/dr+\alpha^2-\lambda^2r^2\) is the Bessel differential operator. Using a comparison of the solutions of the considered problem by the Cauchy function method and by the method of hybrid integral transforms of Fourier-(Kantorovich-Lebedev)-Fourier type the authors derive a representation of a family of improper integrals of hyperbolic functions and of modified Bessel functions.