Logarithmic expansion with cosines
I found the following expansion in this paper:
$$\log\frac{|\boldsymbol{r}-\boldsymbol{r'}|}{L}=\log\frac{r_>}{L}-\sum_{n=1}^{\infty}\frac{1}{n}\left(\frac{r_<}{r_>}\right)^n\cos[n(\phi-\phi')]$$
where $r_>=\max(|\boldsymbol{r}|,|\boldsymbol{r'}|)$, and $r_<=\min(|\boldsymbol{r}|,|\boldsymbol{r'}|)$
Can anyone point me to a paper/reference/explanation of this, please?
$\endgroup$ 11 Answer
$\begingroup$noting the series expansion $\log(1-x)=-\sum_{n=1}^\infty x^n/n$, you see that the right-hand-side of your equation is just
$$\log\frac{r_>}{L}-\sum_{n=1}^{\infty}\frac{1}{n}\left(\frac{r_<}{r_>}\right)^n\cos[n(\phi-\phi')]=\log\frac{r_>}{L}+{\rm Re}\,\log\left(1-\frac{r_<}{r_>} e^{i\phi-i\phi'}\right)$$ $$={\rm Re}\,\log\left(\frac{r_>}{L} -\frac{r_<}{L} e^{i\phi-i\phi'}\right)={\rm Re}\,\log\left(\frac{r_>}{L} e^{i\phi'}-\frac{r_<}{L} e^{i\phi}\right)$$
which obviously [*] equals the left-hand-side, so you're done.
[*] place the vectors $\mathbf{r}$ and $\mathbf{r}'$ in the $x$-$y$ plane, so $x+iy=r_< e^{i\phi}\equiv z$, and $x'+iy'=r_> e^{i\phi'}\equiv z'$ and then note that ${\rm Re}\,\log(z-z')=\log|z-z'|$
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