Int{x=0 to infinity}(exp(-x)*Ln(x)*Ln(x)dx)

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The discussion focuses on evaluating the integral Int{x=0 to infinity}(exp(-x)*Ln(x)*Ln(x)dx). Participants note that while indefinite integrals involve complex hypergeometric and error functions, the definite integrals yield simpler results. Key insights include the relationship to Euler's gamma function, specifically that the first integral equals Gamma''(1) and relates to Euler's Constant, Gamma'(1). The second integral, Int{x=0 to Infinity}(exp(-x*x)*Ln(x)dx), also connects to Gaussian integrals.

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  • Knowledge of hypergeometric functions and error functions
  • Basic concepts of Gaussian integrals
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mathslover
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I have tried many hours on the following integrals and would appreciate any help from you.


1. Int{x=0 to infinity}(exp(-x)*Ln(x)*Ln(x)dx)

2. Int{x=0 to Infinity}(exp(-x*x)*Ln(x)dx)



Any idea guys?
 
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Quickly checked it with Mathematica... the indefinite integrals are no fun (they involve hypergeometric functions and error functions) but the definite integrals come out relatively nicely.

I suppose you will want to somehow use
\int_0^\infty e^{-t} \ln(t) \, \mathrm dt = -\gamma
(the Euler gamma). The second one will also involve a Gaussian integral.

Will think a bit more...
 
This has nothing to do with "number theory". I am moving it to Calculus and Analysis.
 
For the first integral ,

it can be shown to be = Gamma''(1) and

- Euler's Constant = Gamma'(1)

where Gamma(x)=Gamma Integral

I just don't know how to use the above facts.
 

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