Debye functions.

Debye functions.

[Ed Smith-Rowland]

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.

The values returned by gsl debye functions start at one for x=0 and 
monotonically decrease.

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.

Thank you.

Ed Smith-Rowland

Re: Debye functions.

[maxgacode]

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

Re: 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