An exact 2-D integration challenge
- From: "Simon Tyler" <simonjtyler@xxxxxxxxx>
- Date: 14 Aug 2006 03:40:25 -0700
Firstly I appologise for stealing Vladimir's title...
I have a couple of double integrals that I expect (with fair reason)
are possible to do in closed form in terms of polygamma functions.
They are (in mathematica notation and x>0 )
Integrate[ Csch[s+t]^2 (s^2 Csch[s]^2 - 1)Exp[-2x(s+t)],
{s,0,Infinity}, {t,0,Infinity} ]
Integrate[ Csch[s+t]^2 (1-s Csch[s] t Csch[t] Cosh[s+t])Exp[-2x(s+t)],
{s,0,Infinity}, {t,0,Infinity} ]
I've calculated a few similar examples such as
Integrate[s t Csch[s+t]^2 Exp[-2x(s+t)], {s,0,Infinity},{t,0,Infinity}
]
= -1/24 (3 PolyGamma[2,x] + x PolyGamma[3,x] )
(note this can be generalised to any power of s or t...)
and
4x Integrate[Coth[s+t] (s Csch[s]^2 - s^2 Coth[s]Csch[s]^2 )
Exp[-2x(s+t)], {s,0,Infinity},{t,0,Infinity} ]
=1 + 3 x - 4 x^2 PolyGamma[1, x] - x^3 PolyGamma[2, x]
So any solutions or good ideas about how to attack the problem will be
more than welcome,
many thanks for any attempts in advance,
Simon
PS
If anyone has any questions or want to know how I did the above
examples, please do not hesitate to ask.
.
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