Voorhoeve index

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.

The Voorhoeve index ${\displaystyle V_{I}(f)}$ of a complex-valued function f that is analytic in a complex neighbourhood of the real interval ${\displaystyle I}$ = [a, b] is given by

(Different authors use different normalization factors.)

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 ${\displaystyle N_{I}(f)}$ denotes the number of zeros of the continuously differentiable function f on the interval ${\displaystyle I}$, then ${\displaystyle N_{I}(f)}$ ≤ ${\displaystyle N_{I}}$(f ') + 1.

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