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

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Discussion Overview

The discussion revolves around the evaluation of two specific integrals involving logarithmic functions and exponential decay. The focus is on theoretical approaches and mathematical reasoning related to these integrals.

Discussion Character

  • Mathematical reasoning, Technical explanation, Homework-related

Main Points Raised

  • One participant seeks assistance with the integrals: Int{x=0 to infinity}(exp(-x)*Ln(x)*Ln(x)dx) and Int{x=0 to Infinity}(exp(-x*x)*Ln(x)dx).
  • Another participant notes that the indefinite integrals are complex, involving hypergeometric functions and error functions, but suggests that the definite integrals are more manageable.
  • A different participant points out that the first integral can be related to Gamma functions, specifically stating it equals Gamma''(1) and references Euler's Constant as Gamma'(1), but expresses uncertainty about how to apply this information.
  • There is a correction regarding the categorization of the thread, with one participant moving it from "number theory" to "Calculus and Analysis."

Areas of Agreement / Disagreement

Participants express differing views on the complexity of the integrals and how to approach them, with no consensus reached on a definitive method or solution.

Contextual Notes

The discussion includes references to specific mathematical functions and constants, but lacks detailed explanations of the assumptions or definitions involved in the integrals.

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
[tex]\int_0^\infty e^{-t} \ln(t) \, \mathrm dt = -\gamma[/tex]
(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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