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Let $a$ be a complex number. Prove that there exists a power series $$ f(z)=z+c_{2} z^{2}+c_{3} z^{3}+\ldots $$ with a positive convergence radius $R,$ so that $$ f(z)+a f(z)^{2}=z, \quad|z|<R $$ How to prove such an existence? How the coefficients $c_2,~c_3,...$ can be calculated? How can Taylor's theorem be helpful?

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  • $\begingroup$ Is this a Real Analysis question? Or a Complex Analysis one? $\endgroup$ – José Carlos Santos Jan 24 at 9:54
  • $\begingroup$ How is this related to Taylor's theorem? $\endgroup$ – Martin R Jan 24 at 10:25
  • $\begingroup$ Complex analysis $\endgroup$ – RIYASUDHEEN T. K Jan 24 at 10:34
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Since the question is tagged complex-analysis, here is such an argument:
If $a=0$ the statement is trivial, so assume that $a\ne0$. Now $f(z)+af(z)^2=z$ is equivalent to $$\left(f(z)+\frac{1}{2a}\right)^2=\frac{z}{a}+\frac{1}{(2a)^2}$$ Let $h(z)=\frac{z}{a}+\frac{1}{(2a)^2}$. Since $h(0)\ne0$ there is some neighborhood of $0$ in which $h$ has a square root $g$ where we may assume $g(0)=\frac{1}{2a}$. Now set $f(z)=g(z)-\frac{1}{2a}$ and one can check that it satisfies the desired properties.

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