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* Debye functions.
@ 2017-03-24  9:20 Ed Smith-Rowland
2017-03-25 21:24  maxgacode
From: Ed Smith-Rowland @ 2017-03-24  9:20 UTC (permalink / raw)
To: gsl-discuss

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

^ permalink raw reply	[flat|nested] 3+ messages in thread

* Re: Debye functions.
2017-03-24  9:20 Debye functions Ed Smith-Rowland
@ 2017-03-25 21:24  maxgacode
2017-03-27 19:04    Ed Smith-Rowland
From: maxgacode @ 2017-03-25 21:24 UTC (permalink / raw)
To: gsl-discuss

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.

\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

^ permalink raw reply	[flat|nested] 3+ messages in thread

* Re: Debye functions.
2017-03-25 21:24  maxgacode
@ 2017-03-27 19:04    Ed Smith-Rowland
0 siblings, 0 replies; 3+ messages in thread
From: Ed Smith-Rowland @ 2017-03-27 19:04 UTC (permalink / raw)
To: gsl-discuss

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.
>
>
>
> \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

^ permalink raw reply	[flat|nested] 3+ messages in thread

end of thread, other threads:[~2017-03-27 19:04 UTC | newest]

2017-03-25 21:24  maxgacode

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