#### Publication Date

9-12-2014

#### Abstract

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.

#### Degree Name

Mathematics

#### Level of Degree

Masters

#### Department Name

Mathematics & Statistics

#### First Advisor

Lau, Stephen

#### First Committee Member (Chair)

Blair, Matthew

#### Second Committee Member

Ellison, James

#### Project Sponsors

National Science Foundation

#### Language

English

#### Keywords

Macdonald function, Bessel equation, radial wave equation, Newton's identities, Mathematica

#### Document Type

Thesis

#### Recommended Citation

Tejeda, Kaylee. "On Roots of the Macdonald Function." (2014). https://digitalrepository.unm.edu/math_etds/47