On 03/25/2017 05:24 PM, maxgacode wrote:
Il 24/03/2017 10:19, Ed Smith-Rowland ha scritto:
Greetings,
I've been looking at the Debye integrals
D_n(x) = \frac{n}{x^n}\int_{0}^{x} \frac{t^n}{e^t - 1}dt
The integrand is everywhere positive.
The definite integral must be zero for x=0.
But the 1/x factor goes to zero and so you get a 0/0 indeterminate
ratio. Computing the limit to zero returns 1.0!
The values returned by gsl debye functions start at one for x=0 and
monotonically decrease.
Please note the factor
\frac{n}{x^n}
That factor is the responsible of the observed behavior.
The definite integral of a positive functions must start at zero and
monotonically increase.
Is it possible that we have a complementary Debye integral? Perhaps
scaled?
In any case, the functions can't match the formulas in the manual.
I don't think so. Please try to multiply the result of
gsl_sf_debye_n(x) by n/x^n and see.
Moreover the Chapter 27 of Abramowitz and Stegun (page 998 of my ninth
edition) is listing the values of the Debye functions, you can easily
verify that GSL implementation is correct.
Hope this helps
Max
Ah. This is just a convention. Wolfram and others lose the n/x^n.
So the thins look sigmoid and level off at \Gamma(n+1)\zeta(n+1).
Sorry for the noise.
Ed
