Conical functions of purely imaginary order and argument (Q2871077)

The author constructs asymptotic expansions of the solutions to the associated Legendre equation NEWLINE\[NEWLINE\displaystyle(1+x^2)y_{xx}+2xy_x+\bigl(\tau^2+\frac{1}{4}-\frac{\mu^2}{1+x^2}\bigr)y=0NEWLINE\]NEWLINE with respect to the large real parameters \(\tau\) and \(\mu\) while the independent variable \(x\) belongs to an infinite interval of the real line. The work is motivated by applications of such expansions to the theory of the Laplace-Beltrami operator on compact hyperbolic Riemann surfaces.NEWLINENEWLINEThe asymptotic expansions are obtained by using the conventional Liouville-Green (WKBJ) approach enhanced by a smart use of the changes of the dependent and independent variables. The location of two existing transition points with respect to each other and to the singular points suggests the convenient choice of one or another transcendent function involved in the asymptotic expansion. The latter is constructed in the form of the sum of the above transcendent function and its derivative each multiplied by a finite sum in inverse degrees of the large parameter. Coefficients to the latter finite sums are determined recursively using certain integral expressions. The author finds also the \(x\)-dependent bounds to the relevant error terms in the asymptotic approximations.