Re: Re: limit
- To: mathgroup at smc.vnet.net
- Subject: [mg78621] Re: [mg78545] Re: limit
- From: DrMajorBob <drmajorbob at bigfoot.com>
- Date: Thu, 5 Jul 2007 04:12:48 -0400 (EDT)
- References: <f6d9si$2l5$1@smc.vnet.net> <19518154.1183545117387.JavaMail.root@m35>
- Reply-to: drmajorbob at bigfoot.com
That seems plausible, of course, but numerical results contradict it, both
for the original expression and the denominator derivative.
Start by taking apart the OP's expression:
Clear[expr, noProblem, zero, infinite]
expr[s_] = -((2^(-2 + s)*Cos[(1/4)*Pi*(1 + s)]*Gamma[(1 + s)/4]^2*
Gamma[(1 +
s)/2]*
HypergeometricPFQ[{1/4 + s/4, 1/4 + s/4, 1/4 + s/4,
3/4 + s/4},
{1/2, 1, 1}, 1])/Pi);
List @@ % /. s -> 1
{-1, 1/2, 1/\[Pi], 0, \[Pi], 1, \[Infinity]}
noProblem[s_] = expr[s][[{1, 2, 3, 5, 6}]]
zero[s_] = expr[s][[4]]
infinite[s_] = expr[s][[-1]]
-(2^(-2 + s) Gamma[(1 + s)/4]^2 Gamma[(1 + s)/2])/\[Pi]
Cos[1/4 \[Pi] (1 + s)]
HypergeometricPFQ[{1/4 + s/4, 1/4 + s/4, 1/4 + s/4, 3/4 + s/4}, {1/2,
1, 1}, 1]
Here's essentially the same result you have:
D[zero[s], s] /. s -> 1
D[1/infinite[s], s] /. s -> 1
-\[Pi]/4
0
But the latter derivative is clearly wrong (probably due to some
simplification rule, applied without our knowledge, that isn't correct at
s=1):
Clear[dy, infInverse]
dy[f_, dx_, digits_] := N[(f[1 - dx] - f[1 - 2 dx])/dx, digits]
infInverse[s_] = 1/infinite[s];
Table[dy[infInverse, 10^-k, 25], {k, 1, 10}]
{-1.923427116516395532881478, -2.983463609047004067632190, \
-3.125313973445201143419500, -3.139959997528752908922902, \
-3.141429339969442289280062, -3.141576321747481534963308, \
-3.141591020400759167851555, -3.141592490270841802264744, \
-3.141592637257897614551351, -3.141592651956603671268599}
Table[Pi + dy[infInverse, 10^-k, 30], {k, 10, 20}]
{1.63318956719404435692*10^-9, 1.6331895676743358738*10^-10,
1.633189567722365025*10^-11, 1.63318956772716794*10^-12,
1.6331895677276482*10^-13, 1.633189567727696*10^-14,
1.63318956772770*10^-15, 1.6331895677277*10^-16,
1.633189567728*10^-17, 1.63318956773*10^-18, 1.6331895677*10^-19}
So the denominator limit is (almost certainly) -Pi, and the original limit
is
noProblem[1] D[zero[s], s]/-Pi /. s -> 1
% // N
-1/8
-0.125
which agrees perfectly with the OP's calculations and with these:
Table[N[1/8 + expr[1 - 10^-k], 20], {k, 1, 10}]
{-0.031802368727291105600, -0.0029147499047748859136, \
-0.00028902994360074334672, -0.000028878738171604118632, \
-2.8876314481087565099*10^-6, -2.8876072131304841863*10^-7, \
-2.8876047896519244498*10^-8, -2.8876045473042611497*10^-9, \
-2.8876045230694967464*10^-10, -2.8876045206460203254*10^-11}
I don't know how to prove the result symbolically, however.
Bobby
On Wed, 04 Jul 2007 04:28:36 -0500, dh <dh at metrohm.ch> wrote
>
>
> Hi Dimitris,
>
> I think the limit is -Infinity.
>
> consider the following trick :
>
> f1=2^(-2+s)*Cos[(1/4)*Pi*(1+s)]*Gamma[(1+s)/4]^2*Gamma[(1+s)/2]
>
> f2= 1/HypergeometricPFQ[{1/4 + s/4, 1/4 + s/4, 1/4 + s/4, 3/4 + s/4},
>
> {1/2, 1, 1}, 1]
>
> then we are interessted in the limit of f1/f2. As both these expressions
>
> are 0 for s=1, we can take the quotient of the drivatives:
>
> D[f1,s]=-\[Pi]^2/8
>
> D[f2,s]= 0
>
> therefore we get -Infinity
>
> hope this helps, Daniel
>
>
>
>
>
> dimitris wrote:
>
>> Hello.
>
>>
>
>> Say
>
>>
>
>> In[88]:=
>
>> o = -((2^(-2 + s)*Cos[(1/4)*Pi*(1 + s)]*Gamma[(1 + s)/4]^2*Gamma[(1 +
>
>> s)/2]*
>
>> HypergeometricPFQ[{1/4 + s/4, 1/4 + s/4, 1/4 + s/4, 3/4 + s/4},
>
>> {1/2, 1, 1}, 1])/Pi);
>
>>
>
>> I am interested in the value at (or as s->) 1.
>
>>
>
>> I think there must exist this value (or limit) at s=1.
>
>>
>
>> In[89]:=
>
>> (N[#1, 20] & )[(o /. s -> 1 - #1 & ) /@ Table[10^(-n), {n, 3, 10}]]
>
>>
>
>> Out[89]=
>
>> {-0.12528902994360074335,-0.12502887873817160412,-0.12500288763144810876,-0.\
>
>> 12500028876072131305,-0.12500002887604789652,-0.12500000288760454730, -0.\
>
>> 12500000028876045231,-0.12500000002887604521}
>
>>
>
>> However both
>
>> o/.s->1
>
>> and
>
>> Limit[o,s->1,Direction->1 (*or -1*)]
>
>> does not produce anything.
>
>>
>
>> Note also that
>
>>
>
>> In[93]:=
>
>> 2^(-2 + s)*Cos[(1/4)*Pi*(1 + s)]*Gamma[(1 + s)/4]^2*Gamma[(1 + s)/
>
>> 2] /. s -> 1
>
>> HypergeometricPFQ[{1/4 + s/4, 1/4 + s/4, 1/4 + s/4, 3/4 + s/4}, {1/2,
>
>> 1, 1}, 1] /. s -> 1
>
>>
>
>> Out[93]=
>
>> 0
>
>>
>
>> Out[94]=
>
>> Infinity
>
>>
>
>> So am I right and the limit exist (if yes please show me a way to
>
>> evaluate it) or not
>
>> (in this case explain me why; in either case be kind if I miss
>
>> something fundamental!)
>
>>
>
>> Thanks
>
>> Dimitris
>
>>
>
>>
>
>
>
>
--
DrMajorBob at bigfoot.com