I am a math student trying to wrap my head around complex analysis through self-study. I am using Complex Analysis by Serge Lang, but I find myself struggling with some of his power series manipulations and his representations of power series as polynomials. For instance, given the following theorem/proof (taken from Theorem 6.1, p.76):
Theorem Let $f(T) = a_1 T + higher$ $terms$ be a formal power series with $a_1 \not= 0$. Then there exists a unique power series $g(T)$ such that $f(g(T)) = T$. This power series also satisfies $g(f(T)) = T$.
Proof:
For convenience of notation we write $f(T)$ in the form
$f(T) = a_1T - \sum_{2}^{\infty} a_nT^n$
We seek a power series
$g(T) = \sum_{1}^{\infty} b_nT^n$
such that
$f(g(T)) = T$
The solution to this problem is given by solving the equation in terms of the coefficients of the power series
$a_1g(T) - a_2g(T)^2 - ... = T$
These equations are of the form
$a_1b_n - P_n(a_2, ..., a_n, b_1,..., b_{n - 1}) = 0 \quad \text{and} \quad a_1b_1 = 1 \quad \text{for} \quad n = 1$
where $P_n$ is a polynomial with positive integer coefficients (generalized binomial coefficients)
...
I can follow this all right up until the polynomial representation is used. What I would like to do is follow this type of argument with greater ease. What readings and/or exercises should I do accomplish this?