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Closed sum containg Fresnel integral

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-1

$begingroup$

through my work on the completeness relation of an arbitrary wave function on the the Hilbert space of quantum mechanical infinite square potential, I found the following closed sum:
$$sum _n=1^infty fracleft((-1)^n-1right)^2 Cleft(sqrtnright)^2 n^3=fracpi^24$$
where $C(sqrtn)$ is the cosine Fresnel integral:
$$C(sqrtn)=int_0^sqrtncos(s^2)ds$$
my question is if I can evaluate the integral itself using this closed sum

integration sequences-and-series analysis

share|cite|improve this question

asked 2 days ago

R. UsefR. Usef

83

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  ? What do you mean by "evaluate the integral itself"... the integral is the definition of the Fresnel Integral, it is already evaluated. Do you mean to ask if we can evaluate the sum via the integral?
  $endgroup$
  – Brevan Ellefsen
  2 days ago
  

  
  
  
  


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  $begingroup$
  NO, I mean if i can evaluate the integral by using this sum only
  $endgroup$
  – R. Usef
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Then your question is quite meaningless. The integral is as evaluated as it can be. It has no elementary closed form, which is why the Fresnel Integral is a thing in the first place.
  $endgroup$
  – Brevan Ellefsen
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ok. but note that if i can evaluate it for integers $n$, not generally, another question, is there any way to interchange between the sum and the integral?
  $endgroup$
  – R. Usef
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You actually have the square of an integral, which is more complicated. That being said, your summand is $mathcalO(n^-3)$ while your integrand shouldn't grow faster, at least asymptotically, than $mathcalO(n)$ so interchange should be valid if you are careful
  $endgroup$
  – Brevan Ellefsen
  2 days ago
  
  

  
  
  
  

 | 
show 2 more comments

-1

$begingroup$

through my work on the completeness relation of an arbitrary wave function on the the Hilbert space of quantum mechanical infinite square potential, I found the following closed sum:
$$sum _n=1^infty fracleft((-1)^n-1right)^2 Cleft(sqrtnright)^2 n^3=fracpi^24$$
where $C(sqrtn)$ is the cosine Fresnel integral:
$$C(sqrtn)=int_0^sqrtncos(s^2)ds$$
my question is if I can evaluate the integral itself using this closed sum

integration sequences-and-series analysis

share|cite|improve this question

asked 2 days ago

R. UsefR. Usef

83

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  ? What do you mean by "evaluate the integral itself"... the integral is the definition of the Fresnel Integral, it is already evaluated. Do you mean to ask if we can evaluate the sum via the integral?
  $endgroup$
  – Brevan Ellefsen
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  NO, I mean if i can evaluate the integral by using this sum only
  $endgroup$
  – R. Usef
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Then your question is quite meaningless. The integral is as evaluated as it can be. It has no elementary closed form, which is why the Fresnel Integral is a thing in the first place.
  $endgroup$
  – Brevan Ellefsen
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ok. but note that if i can evaluate it for integers $n$, not generally, another question, is there any way to interchange between the sum and the integral?
  $endgroup$
  – R. Usef
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You actually have the square of an integral, which is more complicated. That being said, your summand is $mathcalO(n^-3)$ while your integrand shouldn't grow faster, at least asymptotically, than $mathcalO(n)$ so interchange should be valid if you are careful
  $endgroup$
  – Brevan Ellefsen
  2 days ago
  
  

  
  
  
  

 | 
show 2 more comments

$begingroup$

through my work on the completeness relation of an arbitrary wave function on the the Hilbert space of quantum mechanical infinite square potential, I found the following closed sum:
$$sum _n=1^infty fracleft((-1)^n-1right)^2 Cleft(sqrtnright)^2 n^3=fracpi^24$$
where $C(sqrtn)$ is the cosine Fresnel integral:
$$C(sqrtn)=int_0^sqrtncos(s^2)ds$$
my question is if I can evaluate the integral itself using this closed sum

integration sequences-and-series analysis

share|cite|improve this question

asked 2 days ago

R. UsefR. Usef

83

$endgroup$

share|cite|improve this question

asked 2 days ago

R. UsefR. Usef

83

share|cite|improve this question

asked 2 days ago

R. UsefR. Usef

83

asked 2 days ago

R. UsefR. Usef

83

asked 2 days ago

R. UsefR. Usef

83