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.
Definition.
The Voorhoeve index formula_1 of a complex-valued function "f" that is analytic in a complex neighbourhood of the real interval formula_2 = ["a", "b"] is given by
 formula_3
Rolle's theorem.
Rolle's theorem states that if formula_4 is a continuously differentiable real-valued function on the real line, and formula_5 formula_6, where formula_7 has a zero strictly between formula_8 and formula_9. Or, more generally, if formula_10 denotes the number of zeros of the continuously differentiable function formula_4 on the interval formula_2, then formula_13
Now one has the analogue of Rolle's theorem:
 formula_14
This leads to bounds on the number of zeros of an analytic function in a complex region.
References.
 ;templatestyles src="Refbegin/styles.css" / ;