Consider a function $f(r,R)$ of two variables, and consider that $R = R_0 + x$ for some constant $R_0$.
I'm looking for a series expansion of
$$1 / |r – R_0 – x|$$
for small $x$. In other words, the expansion of $1/|r – R|$ around a specific $R_0$.
Here, $r$ is yet another variable and not just a parameter. So, if it wasn't for the absolute value, I could just compute the derivatives and set up the Taylor series.
However, I am a bit unsure about how to handle the absolute value situation. Since $r$ is a variable, it's not so easy to say that, e.g. "$r – R_0$ is always larger (smaller) than $x$". While it is true that the resulting function of $r$ will typically be evaluated at values of $r$ close to $R_0$, the sign of the expression can still be both positive or negative.
Now, I guess I can use the fact that $\frac{d}{dx} | x | = \mathrm{sig}(x)$, but then higher terms in the series will involve derivatives of the signum, and I don't really want to deal with them…
Is there a nicer way to handle these things? If the variables were vectors in 3D, I could use the multipole expansion, but that doesn't really apply here, does it?
Best Answer
I don't think that the question got a satisfactory answer yet (mostly because of the way in which it was worded).
The first important point is that a Taylor series cannot get around a singularity: it only represents the original function in some open set where the function is smooth. So the distributional derivatives of $\mathrm{sgn}$ are irrelevant: we can only work in the region where the denominator does not vanish.
Second, the talk about two variables is confusing. Let's introduce $t=r-R$. Now we want to expand $1/|t|$ into a series around $t_0=r-R_0$. Here $t_0$ can be positive or negative, but it cannot be zero. The Taylor series $$\frac{1}{|t|}=\frac{1}{|t_0|}\frac{1}{1+t_0^{-1}(t-t_0)}=\frac{1}{|t_0|}\sum_{n=0}^\infty (-1)^n(t_0)^{-n}(t-t_0)^n$$ converges in the neighborhood $\{t:|t-t_0|<|t_0|\}$.