On 5/7/12 1:51 AM, Bart Vandewoestyne wrote:
> On 05/04/2012 05:25 PM, Robert Dodier wrote:
>> On 2012-05-04, Bart Vandewoestyne<Bart.Vandewoestyne at telenet.be> wrote:
>>
>>> The real reason why I wanted to look at the Ge(s) function is that I
>>> want to understand its behavior for s->0. If I plot its modulus, i see
>>> a quite smooth behavior between 1e-7 and 1e-1, but for values lower than
>>> 1e-7 the abs(Ge(s)) function seems to go up again???
>>
>> Try replacing the ordinary floats (approx. 16 digits) with bigfloats
>> (variable number of digits). I replaced the floats with bigfloats in
>> your script as shown by the diff in the PS. Increasing fpprec (50, 100,
>> 200) yields results that seem to converge to approximately 2.3b-4 as s
>> goes to 0. (Usual disclaimer about numerical results applies.)
>>
>> Note that if you change fpprec, you probably have to reconstruct Ge and
>> Gm because some arithmetic has already been carried out by the time you
>> get around to defining the functions.
>>
>> By the way, after replacing the floats, you can still apply plot2d to
>> the function, but since the plotting code wants floats, it's not
>> an entirely accurate representation of the function. So to investigate
>> the behavior close to s = 0, I think you can try something like:
>>
>> foo(s):=abs(Ge(s))$
>> for n thru 16 do print(bfloat(foo(1.0b0/10^n)))$
>>
>> Hope this helps, it's an interesting problem.
>>
>> Robert Dodier
>
> Hmmm... interesting... so according to Robert, the limit for s->0 is
>
> 2.3b-4 (a constant)
>
> while Raymond Toy's conclusion in another reply appears to be approximately
>
> (4.5e-23+%i*3.8e-24)/s (1/s behavior)
>
> Note the dramatic difference between these two! Which of these two is
> closest to the real truth (close to 0, the Ge(s) behaves as a constant
> OR as a 1/s singularity) is *exactly* what I'm interested in :-)
I don't know what the true answer is, but if you use your script and
plot from 1e-11 instead of 1e-8, you'll get a straight line near 1e-11,
so I think that hints that Ge(s) behaves like 1/s.
>
> I didn't have the courage yet to calculate this limit with pencil and
> paper... but looking at the above results, it seems like i might go and
> grab lots of ink and lots of sheets of paper to come to a real analytic
> solution ;-)
Shouldn't be too hard (says someone who hasn't tried it!). The
sqrt(s^2-a) terms are easily approximated near s=0 and coth(x) is
approximately 1/x, so all the terms should be easy to write out.
Ray