Orthogonality, Lommel integrals and cross product zeros of linear combinations of Bessel functions

Abstract

The cylindrical Bessel differential equation and the spherical Bessel differential equation in the interval with Neumann boundary conditions are considered. The eigenfunctions are linear combinations of the Bessel function or linear combinations of the spherical Bessel functions . The orthogonality relations with analytical expressions for the normalization constant are given. Explicit expressions for the Lommel integrals in terms of Lommel functions are derived. The cross product zeros and are considered in the complex plane for real as well as complex values of the index and approximations for the exceptional zero are obtained. A numerical scheme based on the discretization of the two-dimensional and three-dimensional Laplace operator with Neumann boundary conditions is presented. Explicit representations of the radial part of the Laplace operator in form of a tridiagonal matrix allow the simple computation of the cross product zeros.

Microstructural Analysis of Peripheral Lung Tissue through CPMG Inter-Echo Time R2 Dispersion

“…Consequently, the expansion parameters, κ n , have to satisfy the eigenvalue Eq (15) in the limit ρ → ∞, and we find This transcendental equation has to be solved numerically; for large η , the eigenvalues approach infinity with eigenvalue κ 1 ascending the slowest as shown in Fig 2c . The first eigenvalue, κ 1 , can be approximated with Eq (17) as In addition, by solving Eq (9) with Eqs ( 16 ) and ( 1 ), and using analytical techniques from [ 30 ], the dimensionless expansion coefficients, G n , are given as: An expression for G n in terms of trigonometric functions is provided in Eq (28) in Appendix A . Since the eigenvalues κ n depend on the volume fraction η only, the same dependence holds for the expansion coefficients G n and is visualized in Fig 2d .…”

“…[ 26 ]. Our analysis is based on results in [ 29 ], but goes beyond this previous study by providing new expressions for relaxation rate, expansion coefficients and limiting cases by utilizing general boundary conditions and novel analytical techniques established in [ 30 ] for the context of lung tissue imaging, as well as an analysis of the relaxation rate curve inflection.…”

“…The phenomenon of the existence of an exceptional zero of the defining eigenvalue equation has been studied in detail by Gottlieb [ 52 ] and Ziener et al . [ 30 ]. Another contribution to surface relaxation is caused by the immobilization of proton spins after collision with the tissue-air interface, an effect that is comparable to the accelerated relaxation of hydration layers around proteins [ 53 ].…”

“…While, within this context, CPMG signal formation was recently investigated by Ziener et al . [ 29 ], the study at hand extends and furthens this previous analysis by examining relaxation rates through general (Fourier) boundary conditions and provides new and simpler expressions for the relaxation rate and associated coefficients by using novel analytical techniques [ 30 ] to methodologically investigate CPMG signal formation and its relation to microstructural parameters of lung parenchyma and lung-tissue-like hydrogel foams.…”

Lineshape of Magnetic Resonance and its Effects on Free Induction Decay and Steady-State Free Precession Signal Formation

“…e authors declare that they have no con icts of interest. Analytical solutions for this partial di erential equation can be given for di usion in a constant gradient as given in Equation (6) in terms of Airy-functions [32,33], for di usion around a two-dimensional dipole eld [55,56] as given in Equation (7) in terms of Bessel-functions [57] and Mathieufunctions [58]. For dephasing on the alveolar surface as given in Equation (9), the signal evolution can be expressed in terms of spheroidal wave functions [18].…”