[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Show that if

[[math]] h(z) = \frac{1 - \sqrt{1 - 4pqz^2}}{2qz}\ , [[/math]]

then

[[math]] h(1) = \left \{ \begin{array}{ll} p/q, & \mbox{if $p \leq q,$} \\ 1, & \mbox{if \ltmath\gtp \geq q,[[/math]]

} \end{array}\right. </math> and

[[math]] h'(1) = \left \{ \begin{array}{ll} 1/(p - q), & \mbox{if $p \gt q,$}\\ \infty, & \mbox{if \ltmath\gtp = q.[[/math]]

} \end{array}\right. </math>