In mathematics, the Voorhoeve index is a non-negative

real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...

associated with certain functions on the

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

s, named after Marc Voorhoeve. It may be used to extend

Rolle's theorem In calculus, Rolle's theorem or Rolle's lemma essentially states that any real-valued differentiable function that attains equal values at two distinct points must have at least one stationary point somewhere between them—that is, a point wher ...

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

V_I(f)

of a complex-valued function ''f'' that is analytic in a complex

neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural are ...

of the real interval

I

= 'a'', ''b''is given by :

V_I(f) = \frac\int_a^b \! \left,  \frac  \, f(t) \ \,\, dt \, = \frac \int_a^b \! \left,  \left(\frac\right) \ \, dt.

(Different authors use different normalization factors.)

Rolle's theorem

states that if

f

is a

continuously differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...

real-valued function on the

real line In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...

, and

f(a)=

f(b)=0

, where

a, then its derivative f' has a zero strictly between a and b . Or, more generally, if N_I(f) denotes the number of zeros of the continuously differentiable function f on the interval I, then N_I(f) \le N_I(f')+1. Now one has the analogue of Rolle's theorem:

: V_I(f) \le V_I (f') + \frac12. This leads to bounds on the number of zeros of an analytic function in a complex region.

References

* * {{refend Calculus Complex analysis