  If $f(x,k)$ is the Jost solution and $f(x) = f(0,k)$, then the $I$-function
is $I(k) := \frac{f^\prime(0,k)}{f(k)}$. It is proved that $I(k)$ is in
one-to-one correspondence with the scattering triple ${\mathcal S} :=\{S(k),
k_j, s_j, \quad 1 \leq j \leq J\}$ and with the spectral function
$\rho(\lambda)$ of the Sturm-Liouville operator $l= -\frac{d^2}{dx^2} + q(x)$
on $(0, \infty)$ with the Dirichlet condition at $x=0$ and $q(x) \in L_{1,1} :=
\{q: q= \bar q, \int^\infty_0 (1+x) |q(x) dx <
  \infty\}$.
  Analytical methods are given for finding $\mathcal S$ from $I(k)$ and $I(k)$
from $\mathcal S$, and $\rho(\lambda)$ from $I(k)$ and $I(k)$ from
$\rho(\lambda)$. Since the methods for finding $q(x)$ from $\mathcal S$ or from
$\rho(\lambda)$ are known, this yields the methods for finding $q(x)$ from
$I(k)$.
