\contentsline {chapter}{List of Figures}{ix}
\contentsline {chapter}{List of Tables}{x}
\contentsline {chapter}{Glossary}{xi}
\contentsline {chapter}{\numberline {1}Introduction}{1}
\contentsline {chapter}{\numberline {2}Radiation boundary conditions and the radial wave equation}{2}
\contentsline {section}{\numberline {2.1}Overview}{2}
\contentsline {section}{\numberline {2.2}The radial wave equation}{3}
\contentsline {section}{\numberline {2.3}Macdonald function as an outgoing solution}{5}
\contentsline {chapter}{\numberline {3}Nonreflecting Boundary Conditions}{7}
\contentsline {section}{\numberline {3.1}Proof of {\bf Theorem 3.0.1\hbox {}}}{8}
\contentsline {section}{\numberline {3.2}Alternative proof.}{11}
\contentsline {chapter}{\numberline {4}Newton's Identities}{14}
\contentsline {section}{\numberline {4.1}Main result}{14}
\contentsline {section}{\numberline {4.2}Proof of {\bf Theorem 4.1.1\hbox {}}}{16}
\contentsline {chapter}{\numberline {5}Numerical root evaluation and identity check for several values of $\ell $}{19}
\contentsline {section}{\numberline {5.1}Root finding and identity verification for $\ell = 20$ to 27 digits of precision}{20}
\contentsline {chapter}{\numberline {6}Conclusion}{30}
\contentsline {chapter}{\numberline {A}Location and count of Macdonald roots}{31}
\contentsline {section}{\numberline {A.1}Root count for $K_{\ell +1/2}(z)$}{32}
\contentsline {section}{\numberline {A.2}Root count for $K_\nu (z)$}{33}
\contentsline {chapter}{References}{36}