Evaluate Gamma Integral: j,k Positive Constants

[mmzaj]

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 .

 

[JJacquelin]

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.

 

[mathman]

The gamma function ratio is a polynomial in t (degree = j, if j is an integer).

 

[KarmonEuloid]

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?

 

[JJacquelin]

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.

The gamma function ratio is a polynomial in t (degree = j, if j is an integer).

This is a so-called Pochhammer polynomial.