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I have something reminiscent to the renormalization techniques in quantum field theory. A story of infinities cancellation. We can pose the problem as the searching for the potential coming from some field: We have a field in an unidimensional problem $F(x)=1/(x-1)$. We need to calculate the potential difference between $0.5$ and $t$. We have a problem, the divergence at $x=1$, but we know that only the potential differences are physically meaningful.

$$\phi(t)-\phi(0.5)=\int_{0.5}^2\frac{1}{x-1}=\lim_{a \to 0^+}\left(\int_0^{1-a}\frac{1}{x-1}\mathrm dx+\int_{1+a}^t\frac{1}{x-1}\mathrm dx\right)=$$

$$=\lim_{a \to 0^+}\left(-\left[\log(1-x)\right]_{0.5}^{1-a}+\left[\log(x-1)\right]_{1+a}^t\right)=$$

$$=\lim_{a \to 0^+}\left(\log(1-1+a)-\log(0.5)+\log(t-1)-\log(1+a-1)\right)=$$

$$=\log(t-1)-\log0.5$$

Not sure whether or not I understand corrrectly the renormalization thecniques, the elemental-pedagogic examples I've seen have far more ingenuity, but the "hard" part of them is the infinity cancellation with limits.

However, in this forum every time a diverging integral appears, the point is dismissed as not defined no matter whether or not the divergence is between the integral limits.

What's the flaw in my reasoning?

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    I think renormalization is a red herring in this case as it is not what you are doing here. There is a freedom to add any constant to a potential without changing the physics which as you say, translates into only potential differences are physical. Also, have you left out absolute values in your definition of F? – 2017-05-02
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    @user121049 Of course, with absolute value, no cancelation is possible, but I was seeking some example for wich the cancellation were as clear as possible. I've found elementary, but physically realistic, examples that if mine is refuted, those are too. My aim with the question was to get an opinion from mathematicians that I presumed would be in the negative. – 2017-05-02
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    @user121049 http://iopscience.iop.org/article/10.1088/0143-0807/31/2/L02/meta, but there are more I don't dind right now. – 2017-05-02

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