Debye functions.

[Ed Smith-Rowland]

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