Integral Limit for f(z) when H Goes to Infinity

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Discussion Overview

The discussion revolves around the limit of a specific integral involving a function f(z) as the parameter H approaches infinity. Participants explore the implications of different properties of f(z), including its continuity and behavior at infinity, while considering various mathematical techniques such as integration by parts.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant suggests that without knowing the explicit form of f(z), it is impossible to draw conclusions about the limit of the integral.
  • Another participant proposes considering the function f(z) as odd, even, or neither to explore its implications on the integral.
  • Integration by parts is suggested as a method to analyze the integral, with the expectation of obtaining a derivative of f multiplied by a spike function.
  • Concerns are raised about the usefulness of integration by parts, with some participants questioning what can be gained from this approach.
  • There is a suggestion to conduct classical range tests with specific forms of f(z) (constant, logarithmic, exponential) to understand the integral's behavior better.
  • One participant mentions that if the derivative of f vanishes at infinity, the resulting integral after integration by parts may also vanish, leaving only boundary terms to consider.
  • Another participant asserts that under certain assumptions, the boundary term may also approach zero, leading to a potential limit of zero.
  • However, a counterpoint is raised regarding the boundary term, suggesting that the behavior might differ from the previous claim.
  • There is a mention that if f converges to a specific value as z approaches infinity, a particular expression for the limit can be derived.

Areas of Agreement / Disagreement

Participants generally agree that the behavior of f(z) is crucial for determining the limit of the integral, but multiple competing views remain regarding the implications of different properties of f(z) and the effectiveness of various mathematical approaches. The discussion remains unresolved with no consensus on the limit.

Contextual Notes

Limitations include the dependence on the specific form of f(z) and the assumptions regarding its behavior at infinity, which have not been explicitly defined or agreed upon by participants.

traianus
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Hello,
Suppose to have the following integral:

\int \limits _{-H/2}^{+H/2}f(z) \frac{H-2z}{\left[\left(b - y\right)^2 + \left(H/2 - z\right)^2\right]^2}dz

Suppose that f(z) does NOT have a crazy behavior and that does not go to infinity anywhere and that it is continuos. I do not know a priori the expression of f(z).

Now the question: what is the limit of the integral when the parameter H (which appears in the limits and integrand) goes to +\infty ?
 
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In my opinion, it is not possible to say anything without knowing explicitely f(z).
 
That is the issue I have. Let's assume the function to be odd or even or neither.
 
Try integrating by parts once (assuming f is differentiable).
 
If I integrate by parts, what do I gain? I do not see it?
 
You should get the derivative of f time a spike centered at H/2. Pay attention to the boundary terms as well. You can get a nice result if you assume f' is continuous and vanishes at infinity.
 
Last edited:
Nothing concretely can be concluded about convergence or divergence without specifying f(z) I affraid.
Suggestion however:
You might want to try some classical range tests with f(z) being constant ,logarithm and exponential funtions to get the feeling of behaviour.
 
Could you please StatusX post your procedure here? I have the feeling that tehno is right and f(z) must be specified. But we can also ask (this can be useful too) what are the properties of the function f(z) to NOT have infinite limit.
 
All I can really say is that if f' vanishes at infinity, you can show the integral you get after integrating by parts vanishes in the limit. Then you are left with the boundary terms, which should be easy, as long as f(z) asympototes to some limit as z->infinity (which, incidentally, implies the first assumption).
 
  • #10
If I unserstood correctly, when the derivative goes to zero the we have only the boundary term; however, the boundary term goes to zero too because it has H at the denominator. So under the assumptions that StatusX made the limit is zero.
 
  • #11
No, check the boundary term again, the H should drop out.
 
  • #12
If f converges to a point say f_{\infty} then the answer would be
\frac{f_{\infty}}{(b-y)^2}
 
  • #13
Hello balakrishnan_v, could you please post your derivation?
Thank you!
 

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