  Let the map $f:[-1,1]\to[-1,1]$ have a.c.i.m. $\rho$ (absolutely continuous
$f$-invariant measure with respect to Lebesgue). Let $\delta\rho$ be the change
of $\rho$ corresponding to a perturbation $X=\delta f\circ f^{-1}$ of $f$.
Formally we have, for differentiable $A$, $$
\delta\rho(A)=\sum_{n=0}^\infty\int\rho(dx) X(x){d\over dx}A(f^nx) $$ but this
expression does not converge in general. For $f$ real-analytic and Markovian in
the sense of covering $(-1,1)$ $m$ times, and assuming an {\it analytic
expanding} condition, we show that
$$\lambda\mapsto\Psi(\lambda)=\sum_{n=0}^\infty\lambda^n \int\rho(dx)
X(x){d\over dx}A(f^nx) $$ is meromorphic in ${\bf C}$, and has no pole at
$\lambda=1$. We can thus formally write $\delta\rho(A)=\Psi(1)$.
