Show that : int_0^infty e^(-x)/(x^2)(1/(1-e^(-x))-1/x-1/2)^2dx=7/36-ln A+(zeta(3))(2pi^2)

mafalexpicsak 2022-10-16 Answered
Show that :
0 e x x 2 ( 1 1 e x 1 x 1 2 ) 2 d x = 7 36 ln A + ζ ( 3 ) 2 π 2
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Answers (1)

Jovanni Salinas
Answered 2022-10-17 Author has 18 answers
Define
f ( s ) = 0 x s 1 e x x 2 ( 1 1 e x 1 x 1 2 ) 2 d x .
This defines an analytic function on the domain Re ( s ) > 0 and the problem is to evaluate f(1).
We have
f ( s ) = 0 x s 3 e x ( 1 ( 1 e x ) 2 + 1 x 2 + 1 4 2 x ( 1 e x ) 1 1 e x + 1 x ) d x
  = 0 ( x s 3 e x ( e x 1 ) 2 + x s 5 e x + x s 3 e x 4 2 x s 4 e x 1 x s 3 e x 1 + x s 4 e x ) d x .
For Re ( s ) > 4, integrating by parts on the first term gives
f ( s ) = 0 ( ( s 3 ) x s 4 e x 1 + x s 5 e x + x s 3 e x 4 2 x s 4 e x 1 x s 3 e x 1 + x s 4 e x ) d x .
= 0 ( ( s 5 ) x s 4 e x 1 + x s 5 e x + x s 3 e x 4 x s 3 e x 1 + x s 4 e x ) d x .
Again assuming Re(s)>4, this gives us
f ( s ) = ( s 5 ) Γ ( s 3 ) ζ ( s 3 ) + Γ ( s 4 ) + 1 4 Γ ( s 2 ) Γ ( s 2 ) ζ ( s 2 ) + Γ ( s 3 ) ,
We may write
f ( s ) = ( s 4 ) ( s 5 ) ζ ( s 3 ) + 1 + 1 4 ( s 4 ) ( s 3 ) ( s 4 ) ( s 3 ) ζ ( s 2 ) + s 4 ( s 4 ) ( s 3 ) ( s 2 ) ( s 1 ) Γ ( s )
and so
f ( 1 ) = lim s 1 ( ( s 5 ) ζ ( s 3 ) ( s 3 ) ( s 2 ) ( s 1 ) ζ ( s 2 ) + 1 12 ( s 2 ) ( s 1 ) + 1 3 ( s 4 ) ( s 3 ) + s 3 ( s 4 ) ( s 3 ) ( s 2 ) ( s 1 ) )
= lim s 1 ( ( s 5 ) ζ ( s 3 ) ( s 3 ) ( s 2 ) ( s 1 ) ζ ( s 2 ) + 1 12 ( s 2 ) ( s 1 ) + 1 3 ( s 4 ) ( s 2 ) )
= 2 ζ ( 2 ) + ζ ( 1 ) + 1 9
= ζ ( 3 ) 2 π 2 + 1 12 ln A + 1 9
= 7 36 ln A + ζ ( 3 ) 2 π 2 .
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