An overview is given for the Dirichlet-to-Neumann map for outgoing solutions to the radial wave equation' in the context of nonreflecting radiation boundary conditions on a spherical domain. We then consider the Macdonald function K_l+1/2 (z) for l \u2208 Z \u22650 , a solution to the half-integer order modified Bessel equation. This function can be expressed as K_l+1/2 (z) = sqrt(\u03c0/2z)e^−z z^−l p_l (z), where p_l (z) is a degree-l monic polynomial with simple roots in the left-half plane. By exploiting radiation boundary conditions for the 'radial wave equation', we show that the root set of p_l (z) also obeys l additional polynomial constraints. These constraints are in fact Newton's identities which relate a polynomial's coefficients to the power sums of its roots. We follow this with numerical verification up to order l = 20.
Level of Degree
Mathematics & Statistics
First Committee Member (Chair)
Second Committee Member
National Science Foundation
Macdonald function, Bessel equation, radial wave equation, Newton's identities, Mathematica
Tejeda, Kaylee. "On Roots of the Macdonald Function." (2014). https://digitalrepository.unm.edu/math_etds/47