# Consequences of Taylor's theorem [closed]

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 $$f(z)+a f(z)^{2}=z, \quad|z| How to prove such an existence? How the coefficients $$c_2,~c_3,...$$ can be calculated? How can Taylor's theorem be helpful?

• Is this a Real Analysis question? Or a Complex Analysis one? – José Carlos Santos Jan 24 at 9:54
• How is this related to Taylor's theorem? – Martin R Jan 24 at 10:25
• Complex analysis – RIYASUDHEEN T. K Jan 24 at 10:34

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.