Evaluate Gamma Integral: j,k Positive Constants

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Discussion Overview

The discussion revolves around evaluating the integral \(\lim_{T\rightarrow \infty}\frac{1}{2T}\int_{-T}^{T}\frac{\Gamma(3+it)}{\Gamma(3+it-j)}e^{ikt}dt\), where \(j\) and \(k\) are positive constants. The focus includes the convergence of the integral and potential methods for evaluation, such as the Fourier transform.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants express uncertainty about how to evaluate the integral, questioning its convergence as \(T\) approaches infinity.
  • One participant asserts that the integral cannot be evaluated for \(T \to \infty\) due to lack of convergence.
  • Another participant notes that the ratio of gamma functions behaves like a polynomial in \(t\), specifically a Pochhammer polynomial if \(j\) is an integer.
  • There is a suggestion that a Fourier transform might be useful, although this is met with skepticism due to the integral's non-convergence.

Areas of Agreement / Disagreement

Participants generally disagree on the evaluation of the integral, with some asserting it is non-convergent while others explore potential methods for evaluation without reaching consensus.

Contextual Notes

The discussion highlights the integral's dependence on the convergence properties and the nature of the gamma function ratio, which may affect the evaluation approach.

mmzaj
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greetings . any ideas on how to evaluate this integral

\lim_{T\rightarrow \infty}\frac{1}{2T}\int_{-T}^{T}\frac{\Gamma(3+it)}{\Gamma(3+it-j)}e^{ikt}dt

j, k are positive constants .
 
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mmzaj said:
greetings . any ideas on how to evaluate this integral

\lim_{T\rightarrow \infty}\frac{1}{2T}\int_{-T}^{T}\frac{\Gamma(3+it)}{\Gamma(3+it-j)}e^{ikt}dt

j, k are positive constants .

Hi !
this integral cannot be evaluated for T-->infinity because it is not convergent.
 

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The gamma function ratio is a polynomial in t (degree = j, if j is an integer).
 
mmzaj said:
greetings . any ideas on how to evaluate this integral

\lim_{T\rightarrow \infty}\frac{1}{2T}\int_{-T}^{T}\frac{\Gamma(3+it)}{\Gamma(3+it-j)}e^{ikt}dt

j, k are positive constants .

Not sure, but would a Fourier transform help here?
 
KarmonEuloid said:
Not sure, but would a Fourier transform help here?
I don't think so. Since the integral is not convergent for T tending to infinity, the Fourier transform is of no help to find a limit which doesn't exist anyways.

mathman said:
The gamma function ratio is a polynomial in t (degree = j, if j is an integer).
This is a so-called Pochhammer polynomial.
 

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