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

[traianus]

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 $$+\infty$$ ?

 

[quasar987]

In my opinion, it is not possible to say anything without knowing explicitely f(z).

 

[traianus]

That is the issue I have. Let's assume the function to be odd or even or neither.

 

[StatusX]

Try integrating by parts once (assuming f is differentiable).

 

[traianus]

If I integrate by parts, what do I gain? I do not see it?

 

[StatusX]

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.

 

[tehno]

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.

 

[traianus]

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.

 

[StatusX]

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).

 

[traianus]

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.

 

[StatusX]

No, check the boundary term again, the H should drop out.

 

[balakrishnan_v]

If f converges to a point say $$f_{\infty}$$ then the answer would be
$$\frac{f_{\infty}}{(b-y)^2}$$

 

[traianus]

Hello balakrishnan_v, could you please post your derivation?
Thank you!