I am reading Henri Cartan's Elementary Theory of Analytic Functions of One or Several Complex Variables.
The very first chapter of this book is about power series in one variable. In section 9 (pp.26-27) of this chapter, Cartan considers the power series $$\bar{S}(x) =A_1x+\sum_{n=2}^\infty\left(\frac xr\right)^n =A_1x-M\frac{x^2/r^2}{1-x/r}$$ inside the disc of convergence $|x|<r$, where $A_1$ and $M$ are positive constants. From a previous discussion, it is known that as a formal power series, $\bar{S}$ has a compositional inverse $$\bar{T}(x)=\sum_{n=1}^\infty B_nx^n$$ with $B_1>0$ and $B_n\ge0$ such that $\bar{S}\circ\bar{T}=I$ (here $I$ denotes the formal power series $I(x)=x$). Cartan wants to prove that the radius of convergence of $\bar{T}$ is nonzero. He wrote:
We seek, then, a function $\bar{T}(y)$ defined for sufficiently small values of $y$ which is zereo for $y=0$ and which satisfies the equation $\bar{S}(\bar{T}(y))=y$ identically; $\bar{T}(y)$ must satisfies the quadratic equation $$(A_1/r+M/r^2)\bar{T}^2-(A_1+y/r)\bar{T}+y=0,\tag{9.4}$$ which has for [sic.] solution (which vanishes when $y=0$) $$\bar{T}(y)=\frac{A_1+y/r-\sqrt{(A_1)^2-2A_1y/r-4My/r^2+y^2/r^2}}{2(A_1/r+M/r^2)}.$$ When $|y|$ is sufficiently small, the surd is of the form $A_1\sqrt{1+u}$ with $|u|<1$, and so $\bar{T}(y)$ can be expanded as a power series in $y$, which converges for sufficiently small $|y|$. Thus the radius of convergence of this series is $\ne0$, as required.
My questions are two-fold:
- How do we know that $\bar{T}(y)=\dfrac{(\cdots)\color{red}{-}\sqrt{\cdots}}{2(\cdots)}$ but not $\bar{T}(y)=\dfrac{(\cdots)\color{red}{+}\sqrt{\cdots}}{2(\cdots)}$ in the above?
- How do we know that we can pick the sign of the square root consistently when $|y|$ is sufficiently small?
The sign was picked so that $\bar{T}(0)=0$. We need this because $S(0)=0$.