Trigonometric Integrals Research Articles

A remark on the distribution of the zeros of Riemann's zeta function and a continuous analogue of Kakeya's theorem

A certain new symmetric representation of Riemann's xi function is considered. A theorem on the zeros of trigonometric integrals analogous to Kakeya's theorem on the zeros of polynomials with monotonically non-decreasing coefficients is used. A modification of Polya's method is suggested, which allows one to obtain new assertions on the disposition of the zeros of the zeta function.

Kirchhoff approximation for backscattering by partially illuminated circular cylinders: Two-dimensional case

The Kirchhoff approximation is widely used for estimating reflection contributions to the backscattering by rigid surfaces [W. G. Neubauer, J. Acoust. Soc. Am. 35, 279–285 (1963)]. Backscattering of a plane wave by a right-circular cylinder has been approximated this way using Struve and Bessel functions [G. C. Gaunaurd, IEEE J. Ocean Eng. 10, 213–230 (1985)]. For various situations of interest, the cylinder may be partially illuminated such that the illumination is abruptly truncated along lines parallel to the cylinder’s axis. This configuration does not appear to have been analyzed previously. This two-dimensional scattering problem is re-formulated by an application of the plane-wave expansion theorem to a phase factor. The scattering is expressed in terms of an infinite series of terms containing Bessel functions and elementary trigonometric integrals. An evaluation of the integrals and numerical evaluation of the series for the fully illuminated case reduces to the aforementioned Struve and Bessel function result. The partial-illumination case exhibits Fresnel zone behavior and illustrates the limitations of quadratic phase approximations. [Work supported by ONR.]

On Bounds for the Characteristic Functions of Some Degenerate Multidimensional Distributions

Abstract We discuss an application of an inequality for the modulus of the characteristic function of a system of monomials in random variables to the convergence of the density of the corresponding system of the sample mixed moments. Also, we consider the behavior of constants in the inequality for the characteristic function of a trigonometric analogue of the above-mentioned system when the random variables are independent and uniformly distributed. Both inequalities were derived earlier by the author from a multidimensional analogue of Vinogradov's inequality for a trigonometric integral. As a byproduct the lower bound for the spectrum of is obtained, where 𝐴𝑘 is the matrix of coefficients of the first 𝑘 + 1 Chebyshev polynomials of first kind.

Laplace equation and Poisson integral

The resolution of Laplace equation in polar coordinates abuts to Poisson integral. In this paper, it is proved that, the Laplace equation remains invariable by inverse mapping (inversion). After, the Poisson integral is calculated for periodic solution of this equation, for which some trigonometric integrals are calculated on intervals having 2 π for length. Finally two examples are resolved.

Closed form expressions for some series over certain trigonometric integrals

The series (3) and (4), where T(x) denotes trigonometric integrals (2), are represented as series in terms of Riemann zeta and related functions using the sums of the series (5) and (6), whose terms involve one trigonometric function. These series can be brought in closed form in some cases, where closed form means that the series are represented by finite sums of certain integrals. By specifying the function φ(y) appearing in trigonometric integrals (2) we obtain new series for some special types of functions as well as known results.

Some Divergent Trigonometric Integrals

(2001). Some Divergent Trigonometric Integrals. The American Mathematical Monthly: Vol. 108, No. 5, pp. 432-436.

Understanding calculus: a user's guide

Author's Message to the Reader Acknowledgments Lines Parabolas, Ellipses, Hyperbolas Differentiation Differentiation Formulas The Chain Rule Trigonometric Functions Exponential Functions and Logarithms Inverse Functions Derivatives and Graphs Following the Tangent Line The Indefinite Integral The Definite Integral Work, Volume, and Force Parametric Equations Change of Variable Integrating Rational Functions Integration By Parts Trigonometric Integrals Trigonometric Substitution Numerical Integration Limits At Sequences Improper Integrals Series Power Series Taylor Polynomials Taylor Series Separable Differential Equations First-Order Linear Equations Homogeneous Second-Order Linear Equations Nonhomogeneous Second-Order Equations Answers Index About the Author.

Polynomial inequalities on measurable sets and their applications

We study Polya-and Remez-type inequalities for univariate and multivariate polynomials and discuss their applications to Nikolskii-type inequalities and upper estimates of trigonometric integrals.

Error rates for Nakagami-m fading multichannel reception of binary and M-ary signals

This paper derives new closed-form formulas for the error probabilities of single and multichannel communications in Rayleigh and Nakagami-m (1960) fading. Closed-form solutions to three generic trigonometric integrals are presented as part of the main result, providing a unified method for the derivation of exact closed-form average symbol-error probability expressions for binary and M-ary signals with L independent channel diversity reception. Both selection-diversity and maximal-ratio combining (MRC) techniques are considered. The results are generally applicable for arbitrary two-dimensional signal constellations that have polygonal decision regions operating in a slow Nakagami-m fading environments with positive integer fading severity index. MRC with generically correlated fading is also considered. The new expressions are applicable in many cases of practical interest. The closed-form expressions derived for a single channel reception case can be extended to provide an approximation for the error rates of binary and M-ary signals that employ an equal-gain combining diversity receiver.

Some Definite Integrals Associated with the Riemann Zeta Function

The authors aim at deriving a family of series representations for \zeta (2n + 1) (n \in \mathbb N) by evaluating certain trigonometric integrals in several different ways. They also show how the results presented in this paper relate to those that were obtained in other works. Finally, some illustrative computational examples, using Mathematica (Version 4.0) for Linux, are considered.

An Asymptotic Expansion for the Number of Permutations with a Certain Number of Inversions

Let $b(n,k)$ denote the number of permutations of $\{1,\ldots,n\}$ with precisely $k$ inversions. We represent $b(n,k)$ as a real trigonometric integral and then use the method of Laplace to give a complete asymptotic expansion of the integral. Among the consequences, we have a complete asymptotic expansion for $b(n,k)/n!$ for a range of $k$ including the maximum of the $b(n,k)/n!$.

Linearization of Trigonometric Polynomials and Integrals

Linearization of Trigonometric Polynomials and Integrals

Trigonometric Integrals and Hadamard Products

(1999). Trigonometric Integrals and Hadamard Products. The American Mathematical Monthly: Vol. 106, No. 1, pp. 36-42.

Trigonometric Integrals and Hadamard Products

Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsL. R. BraggLOUIS R. BRAGG received his doctorate from the University of Wisconsin-Madison. He has reached the status of professor emeritus of mathematical sciences at Oakland University after having served there as a professor during 1966–1997. His primary research area is partial differential equations with an emphasis on transmutation and complex variable studies. He is also interested in special functions and parametric methods.

Bounds for the Characteristic Functions of the System of Monomials in Random Variables and of Its Trigonometric Analogue

Abstract Using a multidimensional analogue of Vinogradov's inequality for a trigonometric integral, the upper bounds are constructed for the moduli of the characteristic functions both of the system of monomials in components of a random vector with an absolutely continuous distribution in and of the system (cos j 1πξ 1 . . . cos j s πξ s , 0 ≤ j 1, . . . , j s ≤ k, j 1 + . . . + j s ≥ 1), where (ξ 1, . . . , ξ s ) is uniformly distributed in [0; 1] s .

BOUNDS FOR THE CHARACTERISTIC FUNCTIONS OF THE SYSTEM OF MONOMIALS IN RANDOM VARIABLES AND OF ITS TRIGONOMETRIC ANALOGUE

Using a multidimensional analogue of Vinogradov's in- equality for a trigonometric integral, the upper bounds are constructed for the moduli of the characteristic functions both of the system of monomials in components of a random vector with an absolutely con- tinuous distribution in R s and of the system

NORMALIZATION OF THE WAVEFUNCTION FOR THE CALOGERO-SUTHERLAND MODEL WITH INTERNAL DEGREES OF FREEDOM

The exact normalization of a multicomponent generalization of the ground state wave-function of the Calogero-Sutherland model is conjectured. This result is obtained from a conjectured generalization of Selberg’s N-dimensional extension of the Euler beta integral, written as a trigonometric integral. A new proof of the Selberg integral is given, and the method is used to provide a proof of the multicomponent generalization in a special two-component case.

Calculus problems for a new century, edited by Robert Fraga. Pp427. $25. 1993. ISBN 0-88385-084-2 (Mathematical Association of America).

Part I. Calculus: I: Derivatives: 1. Arbitrating disputes: maximizing quadratic polynomials finds a fair solution to a labor dispute 2. Fitting lines to data: minimizing quadratic polynomials finds the regression line to a data set 3. Somewhere within the rainbow: minimizing trigonometric functions explains properties of the rainbow 4. Three optimization problems in computing: maximizing functions shows how to store and transmit data most efficiently 5. Newton's method and fractal patterns: Newton's method generates fractal designs in the complex plane Part II. Calculus I: Differential Equations and Integrals: 6. How old is the Earth? The exponential decay of rubidium is used to date very old rocks 7. Falling raindrops: four differential equation models describe how a raindrop falls 8. Measuring voting power: Integrals of polynomials give a measure of voting power 9. How to tune a radio: trigonometric integrals explain tuning a radio 10. Volumes and hypervolumes: integrals calculate the volumes of objects in three and higher dimensions Part III. Calculus II: 11. Reliability and the cost of guarantees: probability integrals predict how long equipment will last 12. Queueing systems: differential equations and limits analyze waiting lines 13. Moving a planar robot arm: derivatives guide a robot arm along a curve 14. Design curves: parametric curves are used in computer-aided design of automobiles 15. Modeling the AIDS epidemic: differential equations model the spread of AIDS 16. Speedy sorting: integrals, limits, and L'Hopital's rule compare the speed of sorting algorithms Part IV. Calculus of Several Variables: 17. Hydro-turbine optimization: lagrange multipliers allocate water to hydroelectric turbines 18. Portfolio theory: lagrange multipliers choose a best stock-market portfolio.

Generalized localization for the multiple Walsh-Fourier series of functions in ,

In this paper the concept of generalized localization almost everywhere (GL) is analyzed for the multiple Fourier-Walsh series of functions in , , summable over rectangles. (For multiple trigonometric series and Fourier integrals GL was introduced and analyzed earlier by one of the authors.) If , then it is proved that GL holds for double Walsh-Fourier series on any open set. It breaks down on any set which is not dense in if and and also in the class if . All results on the Walsh system obtained in this paper are identical to the results on GL for Fourier series in the trigonometric system obtained earlier by one of the authors.

On three trigonometric integrals of lnΓ( x) or its derivative

Three definite integrals of products of ln Γ( x) or ψ( x) with trigonometric functions have been wrongly believed to be zero for nearly a century and are given in this form even in the most recent tables. A simple result is given for one of these integrals, the result for the two others being expressible in terms of an infinite series.