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\bc
{\Large\bf Numerical Treatment of Spheroidal Wave Functions}

\bigskip

\vspace{0.2cm}

{\large Final Report}

\vspace{0.2cm}

({\it October 30, 1997})
\ec

\bigskip
\bigskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noi {\bf 1$^0$.} The calculations of the light scattering
by a spheroidal particle using the solutions given by the
method of separation of variables are based on the computations
of the spheroidal wave functions$^{1,2}$.

In order to find the scattering characteristics one needs
to calculate a set of the angular spheroidal  functions of first kind
and the radial spheroidal  functions of first and second kinds
$R_{mn}^{(1), (2)}(\alpha c, \alpha^3 \xi)$ of
various indices $m=1, 2, 3,...$ and $n$ ($n = m, m+1, m+2,...$).
Here $\xi$ is the radial coordinate
($1 \leq \xi < \infty$, $\alpha =1$  for the prolate functions and
$0 \leq \xi < \infty$, $\alpha =-i$  for the oblate ones),
$c = 2\pi/\lambda \cdot d/2$ with $d$ being the focal distance
of the spheroid and $\lambda$ being the wavelength of the interacting
wave.

It is well known$^{3-5}$ that the main problem arise in the course of the
computations of functions $R_{mn}^{(2)}(\alpha c, \alpha^3 \xi)$.

\bigskip
\noi {\bf 2$^0$.} The differential equation for the radial spheroidal
functions is given by
\be
\frac{\rm d}{{\rm d}\xi} \left[(\xi^{2} - \alpha^2)
\frac{\rm d}{{\rm d}\xi}R_{mn}^{(1),(2)}(\alpha c, \alpha^3 \xi)\right]
- \left[ \lambda_{mn}(\alpha c)  - c^2 \xi^2 +
\frac{\alpha^2 m^2}{\xi^{2} - \alpha^2} \right ]
R_{mn}^{(1),(2)}(\alpha c, \alpha^3 \xi) = 0,
\ee
where $\lambda_{mn}(\alpha c)$ are  the eigenvalues.

This problem of spheroidal functions is divided into two parts: determination
of the eigenvalues
and the reasonable choice of the method of the spheroidal function expansions.

\bigskip
\noi {\bf 3$^0$.} The expansions in terms of the Legendre  functions
or the Bessel functions are usually used for calculations of the spheroidal
functions
\be
R_{mn}^{(1)}(\alpha c, \alpha^3 \xi) =
\frac{1}{k^{(1)}_{mn}(\alpha c)}
\sum^{\infty}_{r=0,\,1}\!\!\!~^{'}\,
d^{mn}_{r}(\alpha c)P^{m}_{m+r}(\alpha^3 \xi),
\ee
\be
R_{mn}^{(1)'}(\alpha c, \alpha^3 \xi) =
\frac{1}{k^{(1)}_{mn}(\alpha c)}
\sum^{\infty}_{r=0,\,1}\!\!\!~^{'}\,
d^{mn}_{r}(\alpha c)P^{m'}_{m+r}(\alpha^3 \xi),
\ee
\bea
R_{mn}^{(2)}(\alpha c, \alpha^3 \xi) =
\frac{1}{k^{(2)}_{mn}(\alpha c)}
\left[ \sum^{\infty}_{r=-2m,\,-2m+1}\!\!\!\!\!\!\!\!\!\!\!\!\!~^{'}\,
d^{mn}_{r}(\alpha c)Q^{m}_{m+r}(\alpha^3 \xi) +
\right.
&\nonumber \\
\left.
\sum^{\infty}_{r=2m+2,\,2m+1}\!\!\!\!\!\!\!\!\!\!\!\!\!~^{'}\,
d^{mn}_{-r}(\alpha c)P^{m}_{r-m-1}(\alpha^3 \xi) \right] &
\eea
\bea
R_{mn}^{(2)'}(\alpha c, \alpha^3 \xi) =
\frac{1}{k^{(2)}_{mn}(\alpha c)}
\left[ \sum^{\infty}_{r=-2m,\,-2m+1}\!\!\!\!\!\!\!\!\!\!\!\!\!~^{'}\,
d^{mn}_{r}(\alpha c)Q^{m'}_{m+r}(\alpha^3 \xi) +
\right.
&\nonumber \\
\left.
\sum^{\infty}_{r=2m+2,\,2m+1}\!\!\!\!\!\!\!\!\!\!\!\!\!~^{'}\,
d^{mn}_{-r}(\alpha c)P^{m'}_{r-m-1}(\alpha^3 \xi) \right] &
\eea
\bea
R_{mn}^{(1)}(\alpha c, \alpha^3 \xi) =
\left\{ \sum^{\infty}_{r=0,\,1}\!\!\!~^{'}\,
\frac{(2m+r)!}{r!}d^{mn}_{r}(\alpha c) \right \}^{-1}
\left(\frac{\xi^2 - \alpha^2}{\xi^2} \right)^{\frac{m}{2}}
\left(\frac{\pi}{2c \xi} \right)^{\frac{1}{2}} \times
&\nonumber \\
\sum^{\infty}_{r=0,\,1}\!\!\!~^{'}\,
i^{r+m-n}\, \frac{(2m+r)!}{r!} \,
d^{mn}_{r}(\alpha c)\,J_{m+r+\frac{1}{2}}(c\xi),
\eea
\bea
R_{mn}^{(1)'}(\alpha c, \alpha^3 \xi) =
\left\{ \sum^{\infty}_{r=0,\,1}\!\!\!~^{'}\,
\frac{(2m+r)!}{r!}d^{mn}_{r}(\alpha c) \right \}^{-1}
\left(\frac{\xi^2 - \alpha^2}{\xi^2} \right)^{\frac{m}{2}}
\left(\frac{\pi}{2c \xi} \right)^{\frac{1}{2}} \times
&\nonumber \\
\sum^{\infty}_{r=0,\,1}\!\!\!~^{'}\,
i^{r+m-n}\, \frac{(2m+r)!}{r!} \, d^{mn}_{r}(\alpha c)\,
\left\{\left[\frac{m \alpha^2}{\xi(\xi^2-\alpha^2)}
+\frac{m+r}{\xi} J_{m+r+\frac{1}{2}}(c\xi)\right] - c J_{m+r+\frac{3}{2}}(c\xi) \right\},
\eea
\bea
R_{mn}^{(2)}(\alpha c, \alpha^3 \xi) =
\left\{ \sum^{\infty}_{r=0,\,1}\!\!\!~^{'}\,
\frac{(2m+r)!}{r!}d^{mn}_{r}(\alpha c) \right \}^{-1}
\left(\frac{\xi^2 - \alpha^2}{\xi^2} \right)^{\frac{m}{2}}
\left(\frac{\pi}{2c \xi} \right)^{\frac{1}{2}} \times
&\nonumber \\
\sum^{\infty}_{r=0,\,1}\!\!\!~^{'}\,
i^{r+m-n}\, \frac{(2m+r)!}{r!} \,
d^{mn}_{r}(\alpha c)\,Y_{m+r+\frac{1}{2}}(c\xi),
\eea
\bea
R_{mn}^{(2)'}(\alpha c, \alpha^3 \xi) =
\left\{ \sum^{\infty}_{r=0,\,1}\!\!\!~^{'}\,
\frac{(2m+r)!}{r!}d^{mn}_{r}(\alpha c) \right \}^{-1}
\left(\frac{\xi^2 - \alpha^2}{\xi^2} \right)^{\frac{m}{2}}
\left(\frac{\pi}{2c \xi} \right)^{\frac{1}{2}} \times
&\nonumber \\
\sum^{\infty}_{r=0,\,1}\!\!\!~^{'}\,
i^{r+m-n}\, \frac{(2m+r)!}{r!} \, d^{mn}_{r}(\alpha c)\,
\left\{\left[\frac{m \alpha^2}{\xi(\xi^2-\alpha^2)}
+\frac{m+r}{\xi} Y_{m+r+\frac{1}{2}}(c\xi)\right] - c Y_{m+r+\frac{3}{2}}(c\xi) \right\},
\eea
In Eqs.~(2)--(9), $P^{m}_{n}(\alpha^3 \xi)$, $P^{m'}_{n}(\alpha^3 \xi)$
and $Q^{m}_{n}(\alpha^3 \xi)$, $Q^{m'}_{n}(\alpha^3 \xi)$ are the associated
Legendre functions of first and second kinds and their
first derivatives,
$J_{n+\frac{1}{2}}(c\xi)$, $Y_{n+\frac{1}{2}}(c\xi)$
the spherical Bessel functions of
first and second kinds, respectively, $k^{(1)}, k^{(2)}$ constants.
The summation is over either even or odd values of $r$.

\bigskip
\noi {\bf 4$^0$.} Our method of calculations consists of three
independent steps: computations of the eigenvalues,
computations of the expansion coefficients $d^{mn}_{r}$, and the
spherical functions ($P, Q$ or $J, Y$)
or solution of the differential equation (Eq.~(1)).
Formally, the coefficients $d^{mn}_{r}$ do not require in latter case,
however they are always present in the solution to the light scattering
problem by the separation  of variables method$^{1, 2}$.

\medskip

\bitem{}{\bf 4$^0$1.} An exact and effective method for the determination
of the eigenvalues is developed. It includes: a special
procedure to choice an approximate value of $\tilde{\lambda}_{mn}$
and an iterative method to determine its precise value.
The main criterion of the correctness of the results is their
satisfaction of the condition
\be
\lambda_{m,n+1}(\alpha c) - \lambda_{m,n}(\alpha c) \leq \min (c, n).
\ee
The approach developed has been used for the prolate and oblate functions
in order to find the values of the parameter $c$ up to 100
and, evidently, must work for larger values of $c$.

\medskip

\bitem{}{\bf 4$^0$2.} The expansion coefficients $d^{mn}_{r}$
satisfy to the three-term recurrence relations and were
evaluated with the aid of their standard presentation in the form
of infinite fractions$^6$.

\medskip

\bitem{}{\bf 4$^0$3.} The standard methods of calculations of
the Legendre and Bessel functions described
by Erashevskaya et al.$^6$ were applied.
They give good accuracy in the considered range of parameters.

\medskip

\bitem{}{\bf 4$^0$4.} The Runge-Kutta method$^7$
was used for the solution of differential equations.

\medskip

\bitem{}{\bf 4$^0$5.} In order to check the results of calculations
the computed Wronskian
\be
W_{\rm calc} = R_{mn}^{(1)'}(\alpha c, \alpha^3 \xi) R_{mn}^{(2)'}(\alpha c, \alpha^3 \xi) -
R_{mn}^{(1)'}(\alpha c, \alpha^3 \xi) R_{mn}^{(2)}(\alpha c, \alpha^3 \xi)
\ee
was compared with the theoretical one
\be
W_{\rm theor} = \frac{1}{c(\xi^2 - \alpha^2)}.
\ee

\bigskip
\noi {\bf 5$^0$.} The calculations have been performed for the
prolate and oblate radial spheroidal wave functions with the following
parameters: $c=1(1)10(10)100$, $m=1(1)30$, $n=m, m+1, m+2c,...$,
$\xi=1.0001, 1.005(0.05)2.0, 5.0$ (prolate) and
$\xi=0.01,0.1(0.1)1.0, 2.0, 5.0$ (oblate).
The radial coordinate $\xi$ defines
the boundary of a spheroidal particle and is
connected with the particle semiaxes ratio $a/b$
\be
\xi = \frac{a}{b}\left[\left(
            \frac{a}{b}\right)^{2} - 1 \right]^{-\frac{1}{2}}
            \,\,\,\,\,\,\,\,\,\,\, \mbox{for prolate  spheroid,}
\ee
\be
\xi = \left[\left(\frac{a}{b}\right)^{2} - 1 \right]^{-\frac{1}{2}}
            \,\,\,\,\,\,\,\,\,\,\, \mbox{~~~~~~for oblate  spheroid.}
\ee
Some numerical values of $\xi $ are given in Table 1.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[htb]
\bc
\caption{
{\it The values of the radial coordinate $\xi$ for a spheroid
with the semiaxes ratio $a/b$}}
\begin{tabular}{ccc}
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\multicolumn{1}{c}{$~~a/b$} &
\multicolumn{1}{c}{Prolate spheroid} &
\multicolumn{1}{c}{Oblate spheroid} \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
 ~~1.1 & 2.4004 & 2.1822   \\
 ~~1.5 & 1.3416 & 0.8944   \\
 ~~2.0 & 1.1547 & 0.5774   \\
 ~~3.0 & 1.0607 & 0.3536   \\
 ~~4.0 & 1.0328 & 0.2582   \\
 ~~5.0 & 1.0206 & 0.2041   \\
 ~10.0 & 1.0050 & 0.1005   \\
 ~20.0 & 1.0013 & 0.0501   \\
 ~50.0 & 1.0002 & 0.0200   \\
 100.0 & 1.0001 & 0.0100   \\
\noalign{\smallskip}
\hline
\end{tabular}
\ec
\end{table}

The calculations of the radial spheroidal  functions of first and second kinds
$R_{mn}^{(1), (2)}$ were performed using the
expansions in terms of the LEGendre  functions, BESsel functions
and solution of the DIFFerential equations.  Below,
they are denoted as  {  LEG, BES} and {  DIFF}, respectively.
For example, the expression {  BES/DIFF} means that
the functions $R_{mn}^{(1)}$ were calculated
using the Bessel functions and $R_{mn}^{(2)}$
solving the differential equation.
Note that for  calculations of the angular
spheroidal  functions, the expansion in terms of
the Legendre  functions is sufficient in all cases.

\medskip

\bitem{}{{\bf 5$^0$1.} Main results of our calculations of the
oblate functions are presented in Figs.~1--5. The recommendations
are given in Table~2. The figures show the behaviour of the
difference between the calculated and theoretical Wronskians
($\log |W_{\rm calc} - W_{\rm theor}|$). It should be mentioned
that the calculations has been done on PC/AT with double
precision (16 significant digits). From this, it follows that
the values $|W_{\rm calc}-W_{\rm theor}|<10^{-15} \div 10^{-16}$
mean the {\it exact} computations with the accuracy used.
As it seen from the figures, in some cases good results
may be obtained using different methods of calculations.
The solution of differential equation usually
requires slightly larger computational time than other methods but
the distinction is not very pronounced.
\\ Note that the behaviour of the results for intermediate values of $c$ and
$\xi$ and the values of  $m>1$ is similar to those presented in Figs.~1-5.
}

\medskip

\bitem{}{{\bf 5$^0$2.} Figures~6--10 show the  main results of
calculations made for prolate functions. The recommended methods
are given in Table~3.
    }

\bigskip
\noi {\bf 6$^0$.} \underline {The recommended methods} of calculations of
the oblate and prolate radial spheroidal  functions of first and second kinds are summarized
in Tables~2 and 3, respectively.
The nominator gives the method of calculations of $R_{mn}^{(1)}$
and the denominator --- that of $R_{mn}^{(2)}$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[htb]
\bc
\caption{{\it The recommended methods of calculations of the oblate functions
$R_{mn}^{(1)}$ and $R_{mn}^{(2)}$ }}
%\small{
\footnotesize{
\begin{tabular}{ccccc}
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\noalign{\smallskip}
\multicolumn{1}{c}{$~~c$} &
\multicolumn{1}{c}{$a/b \ga 10$} &
\multicolumn{1}{c}{$3<a/b<10$} &
\multicolumn{1}{c}{$1.2 \la a/b \la 3$} &
\multicolumn{1}{c}{$a/b < 1.2$} \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
 $\leq 10$           & LEG/DIFF&  LEG/DIFF & BES/DIFF ($n \leq 30$) & BES/BES  \\
                     & BES/DIFF&  BES/LEG  &DIFF/DIFF ($n \leq 30$) & DIFF/DIFF ($n \la c$)         \\
                     &DIFF/DIFF& DIFF/DIFF & LEG/LEG  ($n > 30$) &          \\
                     &         &  LEG/LEG  & BES/LEG ($n > 30$)  &          \\
\noalign{\smallskip}
 $>10$ \& $\leq 50$  & BES/DIFF& BES/DIFF ($n \la c$) & BES/DIFF ($n \la c$)  & BES/BES \\
                     &DIFF/DIFF& DIFF/DIFF ($n \la c$)& DIFF/DIFF ($n \la c$) & DIFF/DIFF ($n \la c$)        \\
                     &         & LEG/LEG ($n > c$)    & LEG/LEG ($n > c$)     &         \\
                     &         & DIFF/LEG ($n > c$)   & DIFF/LEG ($n > c$)    &         \\
\noalign{\smallskip}
 $>50$               & BES/DIFF& BES/DIFF ($n \la c$) & BES/DIFF ($n \la c$)  & BES/BES \\
                     &DIFF/DIFF& DIFF/DIFF ($n \la c$)& DIFF/DIFF ($n \la c$) & DIFF/DIFF ($n \la c$)        \\
\noalign{\smallskip}
\hline
\end{tabular}
}
\ec
\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[htb]
\bc
\caption{{\it The recommended methods of calculations of the prolate functions
$R_{mn}^{(1)}$ and $R_{mn}^{(2)}$ }}
%\small{
\footnotesize{
\begin{tabular}{ccccc}
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\noalign{\smallskip}
\multicolumn{1}{c}{$~~c$} &
\multicolumn{1}{c}{$a/b \ga 10$} &
\multicolumn{1}{c}{$2<a/b<10$} &
\multicolumn{1}{c}{$1.3 \la a/b \la 2$} &
\multicolumn{1}{c}{$a/b < 1.3$} \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
$\leq 10$           & LEG/LEG ($n \la c$)&  LEG/LEG           & LEG/LEG  & BES/BES ($n \la 1.5c$) \\
                     & LEG/DIFF ($n > c$) &  LEG/DIFF ($n > c$)& LEG/DIFF & BES/LEG ($n > 1.5c$)    \\
                     &                    &                    &          & LEG/LEG ($n > 1.5c$)    \\
\noalign{\smallskip}
 $>10$ \& $\leq 50$  &LEG/LEG ($n \la c$)& LEG/LEG             & BES/BES ($n \la c$)  & BES/BES \\
                     &LEG/DIFF ($n > c$) & LEG/DIFF ($n > c$)  & BES/LEG ($n > c$)    &         \\
                     &                   &                     & LEG/DIFF ($n > c$)   &         \\
\noalign{\smallskip}
 $>50$               &LEG/LEG ($n \la c$)& LEG/LEG           & BES/BES ($n \la c$) & BES/BES \\
                     &LEG/DIFF ($n > c$) & LEG/DIFF ($n > c$)& LEG/LEG ($n > c$)   &         \\
                     &                   &                   & LEG/DIFF ($n > c$)  &         \\
\noalign{\smallskip}
\hline
\end{tabular}
}
\ec
\end{table}

\newpage

\noi{REFERENCES}
\bigskip

\bitem{}1. N.V. Voshchinnikov and V.G. Farafonov,  ``Optical  properties  of
   spheroidal particles," Astrophys. Space Sci. {\bf 204,}
   19-86 (1993).

\bitem{}2. V.G. Farafonov, N.V. Voshchinnikov and V.V. Somsikov,
``Light scattering by a core-mantle spheroidal particle",
 Applied Optics {\bf 35}, 5412-5426  (1996).

\bitem{}3. C. Flammer, {\it Spheroidal  Wave  Functions},
(Stanford Univ. Press, Stanford, 1957).

\bitem{}4. I.V. Komarov, L.I. Ponomarev, and S.Yu. Slavyanov,
   {\it Spheroidal and Coulomb Spheroidal Functions},
   (Nauka, Moscow, 1976).

\bitem{}5. B.P. Sinha, R.H. MacPhie, ``On the computation of the
prolate  spheroid spheroidal radial functions of the second
kind,"  J. Math. Phys. {\bf 16,} 2378-2381  (1975).

\bitem{}6. S.P. Erashevskaya et al.
   {\it Tables of Spheroidal Wave Functions},
   (Nauka i Tekhnika, Minsk, 1973).

\bitem{}7. G.E. Forsythe et al.
   {\it Computer Methods for Mathematical Computations},
   (Prentice-Hill, 1977).

\bigskip
\newpage
\noi{FIGURE CAPTIONS}

\bigskip

Fig. 1. The behaviour of the accuracy of calculations
of the oblate radial spheroidal functions (the difference between
the calculated and theoretical Wronskians) in dependence
on the index $n$. The separate panels show the accuracy
for different combinations of the methods, $\xi=0.1$ and
various values of $c$ = 10 (a), 20 (b), 30 (c),
40 (d), 50 (e) and 100 (f).

\bigskip

Fig. 2. The same as in Fig. 1 but now for $\xi=0.3$.

\bigskip

Fig. 3. The same as in Fig. 1 but now for $\xi=0.6$.

\bigskip

Fig. 4. The same as in Fig. 1 but now for $\xi=0.9$.

\bigskip

Fig. 5. The same as in Fig. 1 but now for $\xi=2.0$.

\bigskip

Fig. 6. The behaviour of the accuracy of calculations
of the prolate radial spheroidal functions (the difference between
the calculated and theoretical Wronskians) in dependence
on the index $n$. The separate panels show the accuracy
for different combinations of the methods, $\xi=1.005$ and
various values of $c$ = 10 (a), 20 (b), 30 (c),
40 (d), 50 (e) and 100 (f).

\bigskip

Fig. 7. The same as in Fig. 1 but now for $\xi=1.05$.

\bigskip

Fig. 8. The same as in Fig. 1 but now for $\xi=1.15$.

\bigskip

Fig. 9. The same as in Fig. 1 but now for $\xi=1.5$.

\bigskip

Fig. 10. The same as in Fig. 1 but now for $\xi=2.0$.

\vfill

\begin{tabular}{ll}
Prof. Dr. Nikolai Voshchinnikov & \hspace{3cm} AIU\\
Representative of the Software  & \hspace{3cm} Friedrich Schiller
University\\
Development Group & \hspace{3cm} \\
\end{tabular}

\end{document}

