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Why is $cos pi x$ needed in $sumlimits_n=-infty^infty g(n)=frac12ioint_=inftyfraccos pi xsin pi xg(x),mathrm dx$?

3

$begingroup$

I am trying to prove this sum formula:$$sum_n=-infty^infty g(n)=frac12ioint_=inftyfraccos pi xsin pi xg(x),mathrm dx.$$I do not understand why $cos pi x$ is needed here. This is because poles of $sin pi x$ are at $x=0,pm 1,pm 2,cdots$

Any help will be highly appreciated.

sequences-and-series complex-analysis contour-integration

share|cite|improve this question

edited Mar 26 at 6:26

J. M. is a poor mathematician

61.3k5152291

asked Dec 6 '18 at 2:30

Hazoor ImranHazoor Imran

161

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What are you integrating over?
  $endgroup$
  – greelious
  Dec 6 '18 at 2:38
  

  
  
  
  


• 
  
  
  

  
  4
  

  
  
  
  
  
  

  
  $begingroup$
  Up to issue how should one take the limit, one has $$frac1sinpi x = sumlimits_k=-infty^infty frac1(x-k)(sin(pi x)') = sumlimits_k=-infty^infty frac(-1)^kpi(x-k)$$ The $cos(pi x)$ factor is there to get rid of the $(-1)^k$ factor in the residues.
  $endgroup$
  – achille hui
  Dec 6 '18 at 3:01
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @user302797 Using unicode to replace correct mathJAX markup is not acceptable. As the system won't know how to replace the unicode on different rendering engine and expression may not scale/style properly.
  $endgroup$
  – achille hui
  Dec 6 '18 at 3:15
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  It is better to use MathJax to encode mathematical characters. It is true that most browsers can render unicode, however, if the characters are encoded using MathJax, MathJax can manage effects like styling and sizing better.
  $endgroup$
  – robjohn♦
  Dec 6 '18 at 5:29
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Note that in largepiπ: $largepiπ$, the first is italic, which is in keeping with the other characters rendered in MathJax, like $abc$. Roman characters are usually used for operator names like $sin(x)$ and $log(x)$. See this meta question
  $endgroup$
  – robjohn♦
  Dec 6 '18 at 5:35
  
  

  
  
  
  

 | 
show 7 more comments

3

$begingroup$

I am trying to prove this sum formula:$$sum_n=-infty^infty g(n)=frac12ioint_=inftyfraccos pi xsin pi xg(x),mathrm dx.$$I do not understand why $cos pi x$ is needed here. This is because poles of $sin pi x$ are at $x=0,pm 1,pm 2,cdots$

Any help will be highly appreciated.

sequences-and-series complex-analysis contour-integration

share|cite|improve this question

edited Mar 26 at 6:26

J. M. is a poor mathematician

61.3k5152291

asked Dec 6 '18 at 2:30

Hazoor ImranHazoor Imran

161

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What are you integrating over?
  $endgroup$
  – greelious
  Dec 6 '18 at 2:38
  

  
  
  
  


• 
  
  
  

  
  4
  

  
  
  
  
  
  

  
  $begingroup$
  Up to issue how should one take the limit, one has $$frac1sinpi x = sumlimits_k=-infty^infty frac1(x-k)(sin(pi x)') = sumlimits_k=-infty^infty frac(-1)^kpi(x-k)$$ The $cos(pi x)$ factor is there to get rid of the $(-1)^k$ factor in the residues.
  $endgroup$
  – achille hui
  Dec 6 '18 at 3:01
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @user302797 Using unicode to replace correct mathJAX markup is not acceptable. As the system won't know how to replace the unicode on different rendering engine and expression may not scale/style properly.
  $endgroup$
  – achille hui
  Dec 6 '18 at 3:15
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  It is better to use MathJax to encode mathematical characters. It is true that most browsers can render unicode, however, if the characters are encoded using MathJax, MathJax can manage effects like styling and sizing better.
  $endgroup$
  – robjohn♦
  Dec 6 '18 at 5:29
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Note that in largepiπ: $largepiπ$, the first is italic, which is in keeping with the other characters rendered in MathJax, like $abc$. Roman characters are usually used for operator names like $sin(x)$ and $log(x)$. See this meta question
  $endgroup$
  – robjohn♦
  Dec 6 '18 at 5:35
  
  

  
  
  
  

 | 
show 7 more comments

$begingroup$

I am trying to prove this sum formula:$$sum_n=-infty^infty g(n)=frac12ioint_=inftyfraccos pi xsin pi xg(x),mathrm dx.$$I do not understand why $cos pi x$ is needed here. This is because poles of $sin pi x$ are at $x=0,pm 1,pm 2,cdots$

Any help will be highly appreciated.

sequences-and-series complex-analysis contour-integration

share|cite|improve this question

edited Mar 26 at 6:26

J. M. is a poor mathematician

61.3k5152291

asked Dec 6 '18 at 2:30

Hazoor ImranHazoor Imran

161

$endgroup$

share|cite|improve this question

edited Mar 26 at 6:26

J. M. is a poor mathematician

61.3k5152291

asked Dec 6 '18 at 2:30

Hazoor ImranHazoor Imran

161

share|cite|improve this question

edited Mar 26 at 6:26

J. M. is a poor mathematician

61.3k5152291

asked Dec 6 '18 at 2:30

Hazoor ImranHazoor Imran

161

edited Mar 26 at 6:26

J. M. is a poor mathematician

61.3k5152291

edited Mar 26 at 6:26

J. M. is a poor mathematician

61.3k5152291

asked Dec 6 '18 at 2:30

Hazoor ImranHazoor Imran

161

asked Dec 6 '18 at 2:30

Hazoor ImranHazoor Imran

161