Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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
up vote 3 down vote accepted

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 $z^2$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.