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Function: sw_spheroid()
void sw_spheroid(float *re, float *im, float mu, int reset,
float a, int s, int l, int m, float eigenvalues[])
This routine computes the spinweighted spheroidal wave
function
. The arguments are:
 re: Output. The real part of the spinweighted spheroidal
wave function.
 im: Output. The imaginary part of the spinweighted spheroidal
wave function.
 mu: Input. The independent variable, with
being a polar angle, of the spinweighted spheroidal wave function;
.
 reset: Input. A flag that indicates that the function should
reset (
) the internally stored normalization of the
spinweighted spheroidal wave function. The reset flag should be set
if any of the following five arguments are changed between calls; otherwise,
set
so that the routine does not recompute the
normalization.
 a: Input. The dimensionless angular momentum parameter,
, of the Kerr black hole for which the spinweighted
spheroidal wave function is associated.
 s: Input. The integervalued spinweight , which should
be set to for a scalar perturbation (e.g., a scalar field perturbation),
for a vector perturbation (e.g., an electromagnetic field
perturbation), or for a spin two perturbation (e.g., a gravitational
perturbation).
 l: Input. The mode integer
.
 m: Input. The mode integer
.
 eigenvalues: Input. An array, eigenvalues[0..3],
which contains the real and imaginary parts of the eigenvalues
and (see below) as follows:
,
,
,
and
. These may be calculated for
a quasinormal mode using the routine qn_eigenvalues().
The spinweighted spheroidal wave function
is also computed using the method of Leaver [27].
We have adopted the following normalization criteria for the spinweighted
spheroidal wave functions
. First, the angleaveraged
value of the squared modulus of
is unity:
. Second, the complex phase is
partially fixed by the requirement that
is real.
Finally, the sign is set to be
for the real part
in the limit that
in order to agree with the sign of the spinweighted spherical
harmonics
(see [26]).
It is sufficient to compute the spinweighted spheroidal wave functions
with and
because of the following
symmetries [29]:

(8.4.175) 
and

(8.4.176) 
where
.
 Author: Jolien Creighton, jolien@tapir.caltech.edu
Next: Example: spherical program
Up: GRASP Routines: Black hole
Previous: Example: eigenvalues program
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Bruce Allen
20001119