Voorhoeve index

Last updated April 15, 2019

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.

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line. The adjective real in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the transcendental numbers, such as $π$ (3.14159265...). In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more.

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable. The concept of function was formalized at the end of the 19th century in terms of set theory, and this greatly enlarged the domains of application of the concept.

A complex number is a number that can be expressed in the form $a + bi$ , where $a$ and $b$ are real numbers, and $i$ is a solution of the equation $x 2 = -1$ . Because no real number satisfies this equation, $i$ is called an imaginary number. For the complex number $a + bi$ , $a$ is called the real part, and $b$ is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world.

Definition

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

In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not hold generally for real analytic functions. A function is analytic if and only if its Taylor series about x₀ converges to the function in some neighborhood for every x₀ in its domain.

In topology and related areas of mathematics, a neighbourhood is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a point is a set of points containing that point where one can move some amount in any direction away from that point without leaving the set.

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.)

Rolle'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.

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 where the first derivative is zero.

In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set $R$ of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one. It can be thought of as a vector space, a metric space, a topological space, a measure space, or a linear continuum.

Now one has the analogue of Rolle's theorem:

V_{I}(f)\leq V_{I}(f')+{\frac {1}{2}}.

This leads to bounds on the number of zeros of an analytic function in a complex region.

Related Research Articles

In mathematics, the gamma function is one of the extensions of the factorial function with its argument shifted down by 1, to real and complex numbers. Derived by Daniel Bernoulli, if $n$ is a positive integer,

In mathematics, the mean value theorem states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints.

In calculus, Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a k-th order Taylor polynomial. For analytic functions the Taylor polynomials at a given point are finite-order truncations of its Taylor series, which completely determines the function in some neighborhood of the point. It can be thought of as the extension of linear approximation to higher order polynomials, and in the case of k equals 2 is often referred to as a quadratic approximation. The exact content of "Taylor's theorem" is not universally agreed upon. Indeed, there are several versions of it applicable in different situations, and some of them contain explicit estimates on the approximation error of the function by its Taylor polynomial.

In mathematics, the Cauchy integral theorem in complex analysis, named after Augustin-Louis Cauchy, is an important statement about line integrals for holomorphic functions in the complex plane. Essentially, it says that if two different paths connect the same two points, and a function is holomorphic everywhere in between the two paths, then the two path integrals of the function will be the same.

In complex analysis, a discipline within mathematics, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula. From a geometrical perspective, it is a special case of the generalized Stokes' theorem.

In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal and unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis is called an orthonormal basis.

A surface of revolution is a surface in Euclidean space created by rotating a curve around an axis of rotation.

In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, u(t) of a real variable and produces another function of a real variable H(u)(t). This linear operator is given by convolution with the function $:$

In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.

In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum of the square of a function is equal to the sum of the square of its transform. It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was later applied to the Fourier series. It is also known as Rayleigh's energy theorem, or Rayleigh's identity, after John William Strutt, Lord Rayleigh.

In mathematics, there are several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet.

In mathematics, a Dirac comb is a periodic tempered distribution constructed from Dirac delta functions

In mathematics, the Z-function is a function used for studying the Riemann zeta-function along the critical line where the argument is one-half. It is also called the Riemann–Siegel Z-function, the Riemann–Siegel zeta-function, the Hardy function, the Hardy Z-function and the Hardy zeta-function. It can be defined in terms of the Riemann–Siegel theta-function and the Riemann zeta-function by

In calculus, Leibniz's rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form

In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function, then the characteristic function is the Fourier transform of the probability density function. Thus it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the characteristic functions of distributions defined by the weighted sums of random variables.

In the mathematics, and specifically complex analysis, Jensen's formula, introduced by Johan Jensen (1899), relates the average magnitude of an analytic function on a circle with the number of its zeros inside the circle. It forms an important statement in the study of entire functions.

The Sokhotski–Plemelj theorem is a theorem in complex analysis, which helps in evaluating certain integrals. The real-line version of it is often used in physics, although rarely referred to by name. The theorem is named after Julian Sochocki, who proved it in 1868, and Josip Plemelj, who rediscovered it as a main ingredient of his solution of the Riemann-Hilbert problem in 1908.

The fundamental theorem of calculus is a theorem that links the concept of differentiating a function with the concept of integrating a function.

In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear integral are also used; contour integral as well, although that is typically reserved for line integrals in the complex plane.

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

Marc Voorhoeve was a Dutch mathematician who introduced the Voorhoeve index of a complex function in 1976.

In computing, a Digital Object Identifier or DOI is a persistent identifier or handle used to identify objects uniquely, standardized by the International Organization for Standardization (ISO). An implementation of the Handle System, DOIs are in wide use mainly to identify academic, professional, and government information, such as journal articles, research reports and data sets, and official publications though they also have been used to identify other types of information resources, such as commercial videos.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

Voorhoeve index

Contents

Definition

Rolle's theorem

Related Research Articles

References