Confider a function $\rho \in C^2( \mathbb{R}^{+,*} ,\mathbb{R}^{+,*} )$ such that for $r \to 0$, the asymptotic form is $\rho(r) \sim \ln(1/r)$, and define the functional $L(f_1,f_2) \equiv \frac{f_2^2}{f_1^2}$ where $f_1$ and $f_2$ are in $C^2( \mathbb{R}^{+,*}, \mathbb{R})$ . Also define $\rho' \equiv d\rho/dr$.
I am looking for a change of variables $\rho(r) \to X(r)$ that is well behaved and such that $L(\rho, \rho')$ becomes well behaved with that change of variables in the following sense:
I seek a injective function $T \in C^2(\mathbb{R}^{+,*}, \mathbb{R})$ such that $X \equiv T(\rho)$ is in $C^2(\mathbb{R}^{+,*}, \mathbb{R})$, and such that :
- $\lim_{r\to 0} r \mapsto X(r)$ and $\lim_{r\to 0} r \mapsto \frac{dX}{dr}$ exist (and are finite).
- the functional $G(X, X') = L\left(T^{-1}(X),\,\frac{d}{dr}(T^{-1}(X) )\,\right)$ (with $X' \equiv dX/dr$) is defined for $X=0$ and for $X'=0$.
Examples of change of variables which don't work:
If $T(x) = x$, then $G(X,X')= \frac{(X')^2}{X^2}$ which is undefined for $X=0$, and $X' \sim 1/r$ as $r \to 0$.
If $T(x) = \ln(x)$, then $G(X,X')= (X')^2$ is well-defined but as $r \to 0$, , $X \sim \ln(\ln(r)) \to \infty $.
If $T(x) = e^{-x}$, then as $r \to 0$, , $X \sim r $ and $X' \sim 1$ so their limit is well defined. But $G(X,X')= \frac{-(X')^2}{X(ln(X))^4}$ is undefined for $X=0$.
If $T(x) = 1/x$, then $G(X,X')= -(X')^2/X^2$ is not well defined for $X=0$ and $X'=0$ and as $r \to 0$, we have $X \sim 1/\ln(r) \to 0$ but $X' \sim \frac{1}{r(\ln(r))^2}$ does not have a finite limit.
I suspect there is no such change of variables, though I would love to be wrong here! If there isn't one , can we prove it?