Hemispherical Function

$\begin{figure}\begin{center}\BoxedEPSF{HemisphereFunction.epsf}\end{center}\end{figure}$

The hemisphere function is defined as

$\begin{displaymath} H(x,y)=\cases{ \sqrt{a-x^2-y^2} & for $\sqrt{x^2+y^2}\leq a$\cr 0 & for $\sqrt{x^2+y^2}>a$.\cr} \end{displaymath}$

Watson (1966) defines a hemispherical function as a function

which satisfies the Recurrence Relations

$\begin{displaymath} S_{n-1}(z)-S_{n+1}(z)=2S_n'(z) \end{displaymath}$

with

$\begin{displaymath} S_1(z)=-S_0'(z). \end{displaymath}$

References

Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, p. 353, 1966.