Re: limit
- To: mathgroup at smc.vnet.net
- Subject: [mg78545] Re: limit
- From: dh <dh at metrohm.ch>
- Date: Wed, 4 Jul 2007 05:28:36 -0400 (EDT)
- References: <f6d9si$2l5$1@smc.vnet.net>
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
>
>