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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 $S$ 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}

See also Cylinder Function, Cylindrical Function


References

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




© 1996-9 Eric W. Weisstein
1999-05-25