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.
of a complex-valued function f that is analytic in a complex neighbourhood of the real interval
= [a, b] is given by (Different authors use different normalization factors.)
Rolle's theorem states that if
is a continuously differentiable real-valued function on the real line, and
denotes the number of zeros of the continuously differentiable function
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.