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Example: spherical program

The program spherical is an example implementation of the routine sw_spheroid() to compute the standard spin-weighted spherical harmonics [26]. The program also computes these functions using equation (3.1) of [26] for comparison. According to the normalization convention stated in subsection [*], the relationship between the spin-weighted spheroidal harmonics and the spin-weighted spherical harmonics is:

\begin{displaymath}
{}_sY_{\ell m}(\theta,\phi) =
(2\pi)^{-1/2}{}_sS_{\ell m}(\cos\theta)e^{im\phi}
\end{displaymath} (8.5.177)

with $a\omega=0$ and  $A=(\ell-s)(\ell+s+1)$.

To invoke the program, type:

spherical $s$ $\ell$ $m$
for the desired (integer) values of $s$, $\ell$, and $m$ ($\ell\ge\vert s\vert$ and $\vert m\vert\le\ell$). The output is three columns of data: the first column is the independent variable $\mu$ between $-1$ and $+1$, the second column is the value of  $(2\pi)^{-1/2}{}_sS_{\ell m}(\mu)$, and the third column is the value of  ${}_sY_{\ell m}(\mu,0)$ as computed from equation (3.1) of [26]. A comparison of the results produced by the program spherical for $\ell=m=-s=2$ with the exact values of ${}_{-2}Y_{22}(\mu,0)=(5/64\pi)^{1/2}(1+\mu)^2$ is shown in table [*].


Table: A comparison of the values of the spin-weighted spherical harmonic ${}_{-2}Y_{22}(\mu,0)$ calculated by equation (3.1) of Goldberg [26], the values of $(2\pi)^{-1/2}{}_{-2}S_{22}(\mu)$ using routine sw_spheroid(), and the values of the exact result $(5/64\pi)^{1/2}(1+\mu)^2$. The three methods give excellent agreement.
$\mu$ Goldberg sw_spheroid() exact
$-0.99$ $1.576955\times10^{-5}$ $1.576955\times10^{-5}$ $1.576958\times10^{-5}$
$-0.95$ $3.942387\times10^{-4}$ $3.942387\times10^{-4}$ $3.942395\times10^{-4}$
$-0.75$ $9.855968\times10^{-3}$ $9.855967\times10^{-3}$ $9.855986\times10^{-3}$
$-0.55$ $3.193334\times10^{-2}$ $3.193333\times10^{-2}$ $3.193340\times10^{-2}$
$-0.35$ $6.662639\times10^{-2}$ $6.662639\times10^{-2}$ $6.663647\times10^{-2}$
$-0.15$ $1.139351\times10^{-1}$ $1.139351\times10^{-1}$ $1.139352\times10^{-1}$
$+0.15$ $2.085525\times10^{-1}$ $2.085525\times10^{-1}$ $2.085527\times10^{-1}$
$+0.35$ $2.874004\times10^{-1}$ $2.874005\times10^{-1}$ $2.874006\times10^{-1}$
$+0.55$ $3.788640\times10^{-1}$ $3.788639\times10^{-1}$ $3.788641\times10^{-1}$
$+0.75$ $4.829430\times10^{-1}$ $4.829430\times10^{-1}$ $4.829433\times10^{-1}$
$+0.95$ $5.996378\times10^{-1}$ $5.996379\times10^{-1}$ $5.996382\times10^{-1}$
$+0.99$ $6.244906\times10^{-1}$ $6.244906\times10^{-1}$ $6.244911\times10^{-1}$


Includes/spherical.tex

Author: Jolien Creighton, jolien@tapir.caltech.edu


next up previous contents
Next: Example: spheroid program Up: GRASP Routines: Black hole Previous: Function: sw_spheroid()   Contents
Bruce Allen 2000-11-19