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 \( {\displaystyle V_{I}(f)} \) of a complex-valued function f that is analytic in a complex neighbourhood of the real interval I = [a, b] is given by
\( V_I(f) = \frac{1}{2\pi}\int_a^b \! \left| \frac{d}{dt} {\rm Arg} \, f(t) \right| \,\, dt \, = \frac{1}{2\pi} \int_a^b \! \left| {\rm Im}\left(\frac{f'}{f}\right) \right| \, dt.
(Different authors use different normalization factors.)
R
olle'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 \( N_I(f) \)denotes the number of zeros of the continuously differentiable function f on the interval I, then \( N_I(f) \)≤ \( N_I(f ') \) + 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
Voorhoeve, Marc (1976), "On the oscillation of exponential polynomials", Math. Z., 151: 277–294, doi:10.1007/bf01214940
Khovanskii, A.; Yakovenko, S. (1996), "Generalized Rolle theorem in \( R^{n} \) and C", J. Dyn. Control Syst., 2: 103–123, doi:10.1007/bf02259625
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
