12 Jul 2012 02:16

## value of gsl_sf_bessel_i0_scaled at 0

Hi All,

Is there any particular reason that gsl_sf_bessel_i0_scaled(0) returns 1
instead of NaN, or is it a bug? The function has different one-sided limits
at 0 (-1 and 1).

Thanks,
Victor


12 Jul 2012 18:59

### Re: value of gsl_sf_bessel_i0_scaled at 0

After some experiments, I don't think it's a bug. In the documentation (
http://www.gnu.org/software/gsl/manual/html_node/Regular-Modified-Spherical-Bessel-Functions.html),
gsl_sf_bessel_i0_scaled
and gsl_sf_bessel_i0_scaled_e are defined as follows:

These routines compute the scaled regular modified spherical Bessel
function of zeroth order, \exp(-|x|) i_0(x).

where

i_l(x) = \sqrt{\pi/(2x)} I_{l+1/2}(x)

If you use the formula for i_0(x) from the documentation, Wolfram Alpha,
which is based on Mathematica, gives different values (-1 and 1) for one
sided limits:

http://www.wolframalpha.com/input/?i=lim+exp%28-%7Cx%7C%29+sqrt%28pi%2F%282x%29%29+BesselI%5B1%2F2%2C+x%5D+as+x+-%3E+0%2B

However, assuming \sqrt(x) = i \sqrt(-x) for negative x, gives

i_0(x) = \exp(-|x|) sinh(x) / x

and i_0(0) = 1 which is consistent with the GSL implementation.

Best regards,
Victor

On Wed, Jul 11, 2012 at 5:16 PM, victor.zverovich <at> gmail.com <
victor.zverovich <at> gmail.com> wrote:



12 Jul 2012 20:43

### Re: value of gsl_sf_bessel_i0_scaled at 0

Sorry for multiple postings, but I've finally found out the source of the
problem. Contrary to the documentation, the gsl_sf_bessel_i0_scaled
function in GSL actually implements

\exp(-|x|) (\sqrt{\pi}/\sqrt{2x}) I_{1/2}(x)

See
http://www.wolframalpha.com/input/?i=exp%28-%7Cx%7C%29+%28sqrt%28pi%29%2Fsqrt%282x%29%29+BesselI%5B1%2F2%2C+x%5D

and not

\exp(-|x|) \sqrt{\pi/(2x)} I_{1/2}(x)

See
http://www.wolframalpha.com/input/?i=exp%28-%7Cx%7C%29+%28sqrt%28pi+%2F+2x%29%29+BesselI%5B1%2F2%2C+x%5D

Note that these are different, since 1/\sqrt{x} != \sqrt{1/x} for negative
x.

Victor

On Thu, Jul 12, 2012 at 9:59 AM, victor.zverovich <at> gmail.com <
victor.zverovich <at> gmail.com> wrote:

> After some experiments, I don't think it's a bug. In the documentation (
>
http://www.gnu.org/software/gsl/manual/html_node/Regular-Modified-Spherical-Bessel-Functions.html), gsl_sf_bessel_i0_scaled
> and gsl_sf_bessel_i0_scaled_e are defined as follows:
>
>   These routines compute the scaled regular modified spherical Bessel