Re: An exact 1-D integration challenge - 47 - (yum-yum... go and surpass all CASs)

Hi,

I did not realize, that the limit computed had something to do with
this integral.
I would like to offer more direct way to prove that this integral
equals -1/8.

For that one notes that

BesselI[0, z] BesselK[0, z] ==
1/(2 Sqrt[\[Pi]]) MeijerG[{{1/2}, {}}, {{0, 0}, {0}}, z^2]

BesselJ[0, z] BesselY[0, z] == -(1/Sqrt[\[Pi]])
MeijerG[{{}, {1/2}}, {{0, 0}, {0}}, z^2]

Then

In[26]:= Integrate[
z*((1/(2*Sqrt[Pi]))*
MeijerG[{{1/2}, {}}, {{0, 0}, {0}}, z^2])*((-(1/Sqrt[Pi]))*
MeijerG[{{}, {1/2}}, {{0, 0}, {0}}, z^2]),
{z, 0, Infinity}]

Out[26]= -(1/8)

Oleksandr Pavlyk
Special Functions Developer
Wolfram Research


On Jul 6, 11:47 am, Vladimir Bondarenko <v...@xxxxxxxxxxxxxxx> wrote:
Hi Dimitris,

Muach thanks for the link!

http://groups.google.gr/group/comp.soft-sys.math.mathematica/msg/fc80...

Using Mathematica, Oleksandr Pavlyk, Special Functions
Developer, Wolfram Research Inc. based on data reported
by you, proves the nice result we found, -1/8.

Well done Mr.Pavlyk, Mr. Anagnostou, and Mr. Bondarenko! ;)

Have a nice weekend,

Vladimir Bondarenko

VM and GEMM architect
Co-founder, CEO, Mathematical Director

http://www.cybertester.com/ Cyber Tester, LLChttp://maple.bug-list.org/ Maple Bugs Encyclopaediahttp://www.CAS-testing.org/ CAS Testing

On Jul 6, 9:14 am, dimitris <dimmec...@xxxxxxxxx> wrote:

Vladimir Bondarenko :

On Jul 3, 4:24 am, Andreas Dieckmann <adieckm...@xxxxxxx> writes:

AD> the value of your integal as -1/8 !

Precisely.

Could you produce more comments?

Best wishes,

Vladimir Bondarenko

VM and GEMM architect
Co-founder, CEO, Mathematical Director

http://www.cybertester.com/Cyber Tester, LLC
http://maple.bug-list.org/ Maple Bugs Encyclopaedia
http://www.CAS-testing.org/CAS Testing

On Jul 3, 4:24 am, Andreas Dieckmann <adieckm...@xxxxxxx> wrote:
Hi, after I found that
lim(s->1) of hypergeometric4f3({1/4+s/4,1/4+s/4,3/4+s/4},{1/2,1,1},1) =
lim(s->1) of hypergeometric2f1(1/4+s/4,1/4+s/4,1,1)/2 it is easy to get the value of your integal as -1/8 !
Andreas

See here

http://groups.google.gr/group/comp.soft-sys.math.mathematica/browse_t...

Cheers
Dimitris


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