For an even function $f(r)$ that is non-zero at $r=0$ and symmetric around $r=0$, i.e. $f(r)=f(-r)$, I want to be able to approximate the expression
$$ \frac{1}{r}\frac{\partial f}{\partial r} \Bigg\vert_{r=0}. \tag{1} $$
However, since I want the approximation at $r=0$, the problem is that the $1/r$ factor makes it diverge. (The purpose of wanting to do this is ultimately to numerically solve a PDE in radial coordinates using finite differences).
I have seen several times (see for example Eq. 12 of this paper or p. 360 of this paper) that the following approximation should hold at the origin, in order to get around the singularity problem $$ \frac{1}{r}\frac{\partial f}{\partial r}\Bigg\vert_{r=0}\approx \frac{\partial^2f}{\partial r^2}\Bigg\vert_{r=0}, \tag{2} $$ and I would like to see where it comes from.
If I try to derive this approximation by Maclaurin expansion in some small $\delta r$ about the point $r=0$, I get the following up to second order: $$ f(\delta r) \approx f(0) + \delta r \frac{\partial f}{\partial r} \Bigg\vert_{r=0} + \frac{1}{2} \delta r^2\frac{\partial^2 f}{\partial r^2} \Bigg\vert_{r=0} + \,\,\,\,... \tag{3} $$ Rearranging, this becomes $$ \begin{align} \frac{\partial^2 f}{\partial r^2} \Bigg\vert_{r=0} &\approx \frac{2}{\delta r^2}\Bigg( f(\delta r) - f(0) - \delta r \frac{\partial f}{\partial r} \Bigg\vert_{r=0} \Bigg) \tag{4}\\ &= \frac{2}{\delta r}\Bigg( \frac{f(\delta r) - f(0)}{\delta r} - \frac{\partial f}{\partial r} \Bigg\vert_{r=0} \Bigg) \tag{5} \end{align} $$ Now I know that the first derivative is defined as $$ \frac{\partial f}{\partial r}\Bigg\vert_{r=0} = \lim_{\delta r\rightarrow 0}\frac{f(\delta r)-f(0)}{\delta r} \tag{6} $$ so for small $\delta r$ the equation should become $$ \begin{align} \frac{\partial^2 f}{\partial r^2} \Bigg\vert_{r=0} &= \frac{2}{\delta r}\Bigg( \frac{\partial f}{\partial r}\Bigg\vert_{r=0}- \frac{\partial f}{\partial r} \Bigg\vert_{r=0} \Bigg) \tag{7}\\ &= 0, \end{align} $$ which is not the same as the approximation given in Eq. (2). Where have I gone wrong here? How can I arrive successfully at Eq. (2)?
Thanks!
The derivative of an even function is odd and is zero at the origin.
By Taylor on $f'$,
$$f'(r)=rf''(0)+o(r)$$
and
$$\lim_{r\to 0}\frac{f'(r)}r=f''(0).$$
Notice that approximation is understood as
$$\left.\left(\frac1r\frac{\partial f}{\partial r}\right)\right|_{r=0}$$ and not
$$\frac1r\left.\left(\frac{\partial f}{\partial r}\right|_{r=0}\right).$$