Degenerate roots of three transcendental equations involving spherical Bessel functions

Robert L. Pexton, Arno D. Steiger
1979 Mathematics of Computation  
Roots of the three transcendental equations j¡(a\)y¡(\) = )i(\)yi(aX), lxji(x)]'x=art[xy^x)]'x=r¡ = WWxsfjLxyfayi'xaaeav and li^)Yx=0lil[yt(x)]'x:=ß = lii(x)]'x=tl{y,(x)]'x=aß that are degenerate for certain values of the parameter a e (0,1) are presented. The symbols /', and y¡ denote the spherical Bessel functions of the first and second kind. Root degeneracies are discussed for each equation individually as well as for pairs of equations. Only positive roots are considered, since the
more » ... s are invariant under the transformations X -►-X, r\ -► -tj, and /u -► -u. When / = 0, only the third equation has nontrivial roots. These roots are identical with the roots of the first equation for / = 1, i.e. n0n = KXn (n = 1, 2, ...). Various graphs of \m, r¡¡n, and Hm display root-degeneracies as intersections of curves. Accurate values of degenerate roots with the corresponding values of a are exhibited in tables. Roots of the third equation for / = 1(1)15, n = 1(1)30, a = 0.1(0.1)0.7, together
doi:10.1090/s0025-5718-1979-0528056-3 fatcat:btyaqknycravfjd3sjpicyf35y