Does the derivative of a holomorphic function $f(t)$ always vanish at multiple roots of $f$?

Question

Does the derivative of a holomorphic function $f(t)$ always vanish at multiple roots of $f$?

I know its true for polynomials, but is it a general fact known to hold for all transcendental functions?

Answer 1

Yes. Suppose $f$ has a multiple root at $z_0$, so that $$\lim_{z\to z_0} \frac{f(z)}{(z-z_0)^2}$$ exists. Now let $g(z) = f(z)/(z-z_0)^2$, such that the value of $g(z_0)$ is given by the above limit. Then $g$ is holomorphic, and $$f'(z) = g'(z)(z-z_0)^2 + 2g(z)(z-z_0)$$ and so we see that $f'(z_0) = 0$.