Lagrange inversion formula example unclear The following example is from De Bruijn's Asymptotic methods in analysis (page 24). The considered equation is 
$x^t = e^{-x}$
The author wants to transform the equation into the form: $w=z/f(z)$, in order to use the Lagrange inversion formula. So he sets $x=1+z$ and $t^{-1}=w$. And then obtains the equation for $f(z)$ as: $f(z)=-z(1+z)/(\log{(1+z}))$. And then says:

The function $f(z)$ is analytic at $z=0$: $f(z)=-1+c_1z+...$. 
It follows that $x=1-t^{-1}-c_1t^{-2}+...$

My question is: This function $f(z)$ at $z=0$ will be $0/0$. i.e. undefined! So how come he says the function is analytic at $z=0$?

It's a removable singularity; we commonly identify functions with removable singularities with the function which agrees with them on the original domain and with the singularities removed. It's a removable singularity because the numerator and denominator are analytic in a neighborhood of $z=0$ and both scale as $O(z)$ as $z \to 0$.
