# Let $y,z$ be two functions such that $y=zf(y)$ where $f$ is a power series wrt $y$. Then we can find of powers series for any $g(y)$

Let $y,z$ be two functions such that $y=zf(y)$ where $f$ is a power series with respect to $y$. Then we can write any $g(y)$ as a power series of $z$ such that

$g(y)=\sum_{k=1}^{\infty} \frac{1}{k!}\bigg( \big(f(y)^kg^{'}(y)\big)^{(k-1)}\bigg|_{y=0}\bigg)z^k$.

(Here, $^{(k-1)}$, $k-1$-th derivative and $g$ is a function which all derivative exist.)

• Hello - would you explain why this statement matters and whether it is true or false? Mar 13, 2015 at 15:16
• Hi, it is true. For any one who interest: Edouard Goursat, A course in mathematical analysis,1904. Can check section 9, pg.189. Mar 13, 2015 at 15:19
• Thank you. Now why should one be interested in this? Mar 13, 2015 at 15:21
• We can calculate $p-$adic behavior of some function with using this function. This allows us to know it for some nice functions. I used it to calculate but it was complicated in the book I wrote. But the answer shows it is not complicated. Mar 13, 2015 at 15:32