# Intuition Behind Krantz Theorem

The theorem I'm referring to is as follows:

Let $z_0$ be a root of a nonzero holomorphic function $f$ , and let $n$ be the least positive integer such that, the $n$-th derivative of $f$ evaluated at $z_0$ differs from zero. Then the power series of $f$ about $z_0$ begins with the $n$-th term, and $f$ is said to have a root of multiplicity (or “order”) $n$. If $n = 1$, the root is called a simple root.

What is the intuition behind this theorem?

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There was a time when it was easy to get a theorem named after you. –  1015 Mar 31 '13 at 1:04
I do not think this is Krantz's theorem... –  Potato Jul 15 '13 at 4:26
In other words, if $f(z_0)=0$, we can write $f(z)=(z-z_0)^n\cdot g(z)$ for a holomorphic function $g$, and the maximal such $n$ is called the multiplicity of the root. For example $f(z)=\cos(z)-1$ has root $0$ with multiplicity $2$, as $$\cos(z)-1=-\frac{z^2}2+\frac{z^4}{4!}-\frac{z^6}{6!}\pm\dots$$ So we can pull out $z^2$