My problem : I have a function that I want to integrate, in the limit that a parameter goes to zero.(adsbygoogle = window.adsbygoogle || []).push({});

I have a function ##f[x,r]##

I want to compute ##F[r] = \int dx f[x,r]## and then series expand as ##r \rightarrow 0##

This is impossible algebraically for me, but may be possible if I can expand ##f[x,r]## about ##r\rightarrow0## first. But I get a divergent result if I do.

For example:

Assume I have the function ##f[x,r] = \frac{1}{x+r}##

Then $$F[r] = \int_0^1 \frac{1}{x+r} dx = log(\frac{1}{r} + 1)$$ and so the expansion is $$F[r] \approx -log(r) + r -\frac{r^2}{2} + ...$$

which is good. That is what I want.

But if I expand first, the truncation will cause a divergence unless i let it go to infinity:

$$\int_0^1 \frac{1}{x+r} dx \approx\int_0^1 (\frac{1}{x} -\frac{r}{x^2}+\frac{r^2}{x^3} + ...)dx

$$

$$ =\bigl[ log(x) + \frac{r}{x} -\frac{r^2}{2 x^2} + (...)\bigr]_0^1$$

which each term is divergent in the lower limit, unless you resum the series completely.

Are there any strategies for regulating this? Z transforms? Plus distributions? etc?

Thank you for any insight.

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# Puiseux/Taylor Expansion of an Integrand pre-Integration

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