In mathematics, the Voorhoeve index is a non-negative real number associated with certain functions on the complex numbers, named after Marc Voorhoeve. It may be used to extend Rolle's theorem from real functions to complex functions, taking the role that for real functions is played by the number of zeros of the function in an interval.
Contents
Definition
The Voorhoeve index
(Different authors use different normalization factors.)
Rolle's theorem
Rolle's theorem states that if f is a continuously differentiable real-valued function on the real line, and f(a) = f(b) = 0, where a < b, then its derivative f ' must have a zero strictly between a and b. Or, more generally, if
Now one has the analogue of Rolle's theorem:
This leads to bounds on the number of zeros of an analytic function in a complex region.