Integration of mixed function at infinity limit

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Discussion Overview

The discussion revolves around the integration of a mixed function at infinity, specifically in the context of calculating the second virial coefficient from interaction potentials. Participants explore the convergence of the integral and the behavior of the function near zero.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Raymond seeks assistance with integrating a function that does not seem to converge at infinity.
  • Some participants suggest that the integral may be difficult to compute analytically and propose checking the behavior near r = 0 by expanding the numerator as a power series.
  • It is noted that if the parameter rs is not zero, the integral is not convergent, and it can only be computed if rs = 0.
  • Further elaboration indicates that if rs is not zero, the function behaves like c/r² near r = 0, leading to divergence as r approaches 0.
  • Conversely, if rs = 0, the constant c becomes zero, allowing for the next term in the expansion to be integrable, thus resolving the integration issue around r = 0.
  • One participant questions the meaning of "rs," asking whether it represents a product of r and s or a constant parameter, indicating potential ambiguity in the discussion.

Areas of Agreement / Disagreement

Participants express differing views on the convergence of the integral based on the value of rs, with some asserting that it diverges unless rs = 0, while others provide conditions under which the integral may be evaluated. The discussion remains unresolved regarding the implications of the parameter rs.

Contextual Notes

The discussion highlights the dependence on the parameter rs and the behavior of the function near r = 0, which is critical for determining convergence. There are unresolved assumptions regarding the nature of the function and the parameter involved.

raymound
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Hi,

I am trying to calculate second virial coefficient from interaction potentias and I have to Integrate at Infinity level and it seems that Integrate doesn't converge.

can you help me integrate this function. Integrate is attached as an image.

Regards
Raymond
 

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It looks like it would be difficult to integrate analytically. To check its behavior near r = 0, expand the numerator as a power series in r. The first term may be ~ r2, in which case you are OK. However I am not sure - it is messy.
 
If rs is not nul, the integral is not convergent.
So the integral can be computed only if rs=0
 
JJacquelin said:
If rs is not nul, the integral is not convergent.
So the integral can be computed only if rs=0

Why?
 
If rs is not nul, the function to be integrated is equivalent to c/r² close to r=0, where c is a constant (Expand the function around r=0).
The integral of c/r² is divergent for r tending to 0.
if rs=0 then c=0 and one can see from the expansion that the next term is integrable. So, there is no integration problem around r=0 in this particular case of rs=0.
All this concerns the question of convergence around r=0 only.
 
Last edited:
JJacquelin said:
If rs is not nul, the function to be integrated is equivalent to c/r² close to r=0, where c is a constant (Expand the function around r=0).
The integral of c/r² is divergent for r tending to 0.
if rs=0 then c=0 and one can see from the expansion that the next term is integrable. So, there is no integration problem around r=0 in this particular case of rs=0.
All this concerns the question of convergence around r=0 only.

My estimate is that the numerator is ~ r2 near r = 0, unless s = 1.
 
mathman said:
My estimate is that the numerator is ~ r2 near r = 0, unless s = 1.

Does "rs" means r multiplied by s, or does "rs" is the symbol of a parameter (i.e. is constant) ?
In the first case you are right. In the second case I am right.
 

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