## On Roots of the Macdonald Function

Please use this identifier to cite or link to this item: http://hdl.handle.net/1928/24518

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Title

On Roots of the Macdonald Function

Author(s)

Tejeda, Kaylee

Advisor(s)

Lau, Stephen

Committee Member(s)

Blair, Matthew

Ellison, James

Ellison, James

Department

University of New Mexico. Dept. of Mathematics and Statistics

Degree Level

Masters

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 ∈ Z ≥0 , a
solution to the half-integer order modified Bessel equation. This function can be expressed
as K_l+1/2 (z) =
sqrt(π/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.

Date

July 2014

Subject(s)

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

##### Collections

- Mathematics (MS) [22]