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I am reading about renormalisation in QED and I come across the term logarithmic divergence several times. Can somebody explain to me about it in simple terms?

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The term 'logarithmic divergence' is normally used for integrals of the type $$ F(x) = \int_{x_0}^x \frac{1}{\xi}\mathrm d\xi $$ (or possibly of the form $F(x) = \int_{x_0}^x \frac{1}{\xi}f(\xi)\mathrm d\xi$ where $f(\xi)$ approaches some finite limit when $\xi\to\infty$). In these cases, the integral diverges to infinity when $x\to\infty$, but it does this relatively slowly: in fact, as a logarithm, since $$ F(x) \approx \log(x) $$ for the finite cas (or $F(x)\approx F_0 \log(x)+\mathrm{regular}(x)$ if a non-constant $f$ is introduced).

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    $\begingroup$ The case $x_0 \to \infty$ would also be a logarithmic divergence, no? $\endgroup$ – gj255 Nov 9 '16 at 19:57
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    $\begingroup$ @gj255 Yes, obviously, as is the case $x_0\to 0$. $\endgroup$ – Emilio Pisanty Nov 9 '16 at 19:57

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