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Integrals involving Dirichlet kernel and sinc function

0

$begingroup$

I am looking to evaluate the following definite integral involving Dirichlet kernel and sinc function with phase.
beginequationI(d) = int_-infty^infty fracsin[N(k^"-k^')]sin(k^"-k^') fracsin(c(k^"-k^'))c(k^"-k^') fracsin(Nk^')sin(k^') fracsin(ck^')ck^' exp(ik^'d+iphi(k^')) rect Big(frack^'k_0Big) dk^' endequation

Seems to be difficult due to Dirichlet kernel. Thank you in advance.

definite-integrals convolution

share|cite|improve this question

edited Mar 28 at 9:25

user16409

asked Mar 25 at 21:02

user16409user16409

85111

$endgroup$

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  $begingroup$
  No, the Dirichlet kernel cancel and with $int_-infty^infty$ instead of $int_0^k_0$ it would have a closed form.
  $endgroup$
  – reuns
  Mar 25 at 22:13
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Kindly explain how they cancel? btw, I corrected for missing constant in the sinc argument.
  $endgroup$
  – user16409
  Mar 25 at 22:52
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  With your $c$ they don't anymore. Why would you want to evaluate $int_0^k_0$ ? With $int_-infty^infty$ it still has a closed form
  $endgroup$
  – reuns
  Mar 25 at 22:54
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  $int_0^k_0$ ? --> I have optical low pass filter that limits spatial frequencies.
  $endgroup$
  – user16409
  Mar 25 at 23:06
  

  
  
  
  

add a comment | 

0

$begingroup$

I am looking to evaluate the following definite integral involving Dirichlet kernel and sinc function with phase.
beginequationI(d) = int_-infty^infty fracsin[N(k^"-k^')]sin(k^"-k^') fracsin(c(k^"-k^'))c(k^"-k^') fracsin(Nk^')sin(k^') fracsin(ck^')ck^' exp(ik^'d+iphi(k^')) rect Big(frack^'k_0Big) dk^' endequation

Seems to be difficult due to Dirichlet kernel. Thank you in advance.

definite-integrals convolution

share|cite|improve this question

edited Mar 28 at 9:25

user16409

asked Mar 25 at 21:02

user16409user16409

85111

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  No, the Dirichlet kernel cancel and with $int_-infty^infty$ instead of $int_0^k_0$ it would have a closed form.
  $endgroup$
  – reuns
  Mar 25 at 22:13
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Kindly explain how they cancel? btw, I corrected for missing constant in the sinc argument.
  $endgroup$
  – user16409
  Mar 25 at 22:52
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  With your $c$ they don't anymore. Why would you want to evaluate $int_0^k_0$ ? With $int_-infty^infty$ it still has a closed form
  $endgroup$
  – reuns
  Mar 25 at 22:54
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  $int_0^k_0$ ? --> I have optical low pass filter that limits spatial frequencies.
  $endgroup$
  – user16409
  Mar 25 at 23:06
  

  
  
  
  

add a comment | 

$begingroup$

I am looking to evaluate the following definite integral involving Dirichlet kernel and sinc function with phase.
beginequationI(d) = int_-infty^infty fracsin[N(k^"-k^')]sin(k^"-k^') fracsin(c(k^"-k^'))c(k^"-k^') fracsin(Nk^')sin(k^') fracsin(ck^')ck^' exp(ik^'d+iphi(k^')) rect Big(frack^'k_0Big) dk^' endequation

Seems to be difficult due to Dirichlet kernel. Thank you in advance.

definite-integrals convolution

share|cite|improve this question

edited Mar 28 at 9:25

user16409

asked Mar 25 at 21:02

user16409user16409

85111

$endgroup$

share|cite|improve this question

edited Mar 28 at 9:25

user16409

asked Mar 25 at 21:02

user16409user16409

85111

share|cite|improve this question

edited Mar 28 at 9:25

user16409

asked Mar 25 at 21:02

user16409user16409

85111

asked Mar 25 at 21:02

user16409user16409

85111

asked Mar 25 at 21:02

user16409user16409

85111