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Can someone help me to regularize the following divergent integral?

$$ \int_0^{1/2}\, \frac{d x}{x^{3/2} (1-x)^{3/2}} $$


Guys, thank you very much for your answers. Thus if I have understood your procedure, the regularized result of this divergent integral (let's do a trivial case) $$ \int_0^\infty{dx} = \lim_{\Lambda\rightarrow \infty} \int_0^\Lambda{dx}=\lim_{\Lambda\rightarrow \infty}\Lambda - 0 \equiv 0 $$ is zero because one simply remove the divergency and the game is over, right?

Well, I would like to have your opinion about this other regularization I have thought of $$ \int_0^\infty{dx} = \lim_{m\rightarrow\infty} \int_0^m{dx} = \lim_{m\rightarrow\infty} 1+\sum_{n=1}^{m-1} 1 = \lim_{m\rightarrow\infty} 1+\sum_{n=1}^{m-1} {1\over n^0} = 1+\zeta(0)=1-{1\over 2}={1\over 2} $$ where I have used the well-known value $\zeta(0)=-1/2$ of the Riemann $\zeta$-function. I was wondering what can be the physical interpretation of such a (naive, I admit) regularization...but maybe there is none and I am just a crazy physicist :)

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    What do you mean?2012-09-17
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    This integral formally diverges at the lower end. However, I know that there are ways to regularize the integral and assign it a finite value (by throwing away a formally infinite quantity, of course).2012-09-17
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    Well, if that's all you want, pick your favorite number between 0 and 1/2 (not including 0), and throw away the integral from zero to your number, and report the finite value you get by integrating from your number to 1/2. If you want more than this, then you have to know what you want, what conditions your regularization has to satisfy.2012-09-17
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    The first question, and @oen's answer, arose in J. Hadamard's "finite partie" idea, which had a similarly disturbingly ad-hoc appearance, but which he used self-consistently to some advantage. At the same time, I'd agree with Gerry M. that one definitely needs some sense of the "function" of the regularization, since, without constraints, a moderately clever person can arrange to get any answer they want, etc.2012-10-04

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