Cotangent Function Research Articles

Closed-form evaluation of some families of cotangent and cosecant integrals

Several families of integrals involving the cotangent or cosecant function have been evaluated in closed form by making use of elementary arguments. Some of them are expressible in terms of the values of the (generalized) Clausen function and Hurwitz zeta function.

Hidden rays of diffraction for dielectric wedge

Electromagnetic diffraction by a dielectric wedge was analyzed using the method of hidden rays. After ordinary ray‐tracing is terminated in the physical region the usual principle of geometrical optics is also applied to trace the hidden rays in the complementary region, in which original media of background and dielectric are exchanged for each other. The diffraction coefficients are constructed by finite series of cotangent functions, which have one‐to‐one correspondence with not only ordinary rays in the physical region but also hidden rays in the complementary region. The angular period of the cotangent functions is adjusted to satisfy the edge condition at wedge tip. The accuracy of the constructed diffraction coefficients was checked by showing how closely the null‐field condition is satisfied in the complementary region.

Multiple Eisenstein series and multiple cotangent functions

We construct the multiple Eisenstein series and we show a relation to the multiple cotangent function. We calculate a limit value of the multiple Eisenstein series.

Limit values of Eisenstein series and multiple cotangent functions

We show a limit formula for Eisenstein series by using the theory of a multiple cotangent function. The value is expressed simply via the Bernoulli number.

Hidden Rays of Diffraction

The method of hidden rays was developed to provide approximate but accurate analytical solution to the diffraction coefficients of composite wedges. After ordinary ray-tracing in physical regions is terminated, hidden rays are traced only by applying the usual principle of geometrical optics in the complementary regions, which are defined by the formulation of dual integral equations. The diffraction coefficients are constructed routinely by sum of cotangent functions, which have one-to-one correspondence with not only the ordinary but also hidden rays. The angular period of the cotangent functions is adjusted to satisfy the edge condition at wedge tip. Compared to the physical optics diffraction coefficients, the presented diffraction coefficients show smooth transition from high to low dielectric constant cases due to total reflections on the complementary dielectric interface. The accuracy of the presented diffraction coefficients is assured by showing how closely the null field condition is satisfied in the complementary region

Applying differential transformation method to the one-dimensional planar Bratu problem

This paper is application of differential transformation method (DTM) to solve Bratu A considerable research works have been conducted recently in applying DTM to different types of partial differential equation of Abdel-Halim Hassan (Chaos, Solitons & Fractals [6]) and fractional differential equations of Arikoglu and Ozkol (Chaos, Solitons & Fractals [1]). The nonlinear eigenvalue problem 0 = + Δ u e u λ in unit square with u = 0 on boundary 1494 I. H. Abdel-Halim Hassan and Vedat Suat Erturk is often referred to as the Bratu problem. The Bratu problem in one-dimensional planar coordinates 0 = + ′ ′ u e u λ with u(0) = u(1) = 0 has two known, bifurcated solutions for values of c λ λ and a unique solution when c λ λ = . The value of c λ is related to fixed point of hyperbolic cotangent function. Two special cases of problem are illustrated by using technique and numerical results and conclusions will be presented. Mathematics Subject Classification: 74S30, 34B15

A generalized coth-function method for exact solution of differential–difference equation

In this paper we propose a new algebraic algorithm to compute the exact travelling wave solutions of nonlinear differential–difference equations. The idea of the proposed algorithm is generated from generalizing the hyperbolic cotangent (coth-) function. Numerical examples are given to illustrate the efficiency and effectiveness of the proposed method. It has been shown that new types of exact travelling wave solutions for these nonlinear differential–difference equations can be constructed.

Virtual Ray-Tracing in Composite Wedge and Constructing the Diffraction Coefficients

To correct the error posed in the PO(physical optics) diffraction coefficients of a composite wedge composed of perfect conductor and lossless dielectric, virtual rays in the complementary region are traced until all of the virtual rays should be located only in the complementary region. The one-to-one correspondence between geometrical rays and cotangent functions provides the extended diffraction coefficients routinely if those angular period is adjusted to satisfy the edge condition at the tip of the composite wedge. It is assured that the extended diffraction coefficients satisfy the null-field condition in t complementary regions quite well.

Inequalities of some trigonometric functions

By using two identities and two inequalities relating to Bernoulli’s and Euler’s numbers and power series expansions of cotangent function, secant function, cosecant function and logarithms of functions involving sine function, cosine function and tangent function, six inequalities involving tangent function, cotangent function, sine function, secant function and cosecant function are established.

Bergman kernels for rectangular domains and multiperiodic functions in Clifford analysis

AbstractIn this paper, we consider rectangular domains in real Euclidean spaces of dimension at least 2, where the sides can be finite, semi‐infinite, or fully infinite. The Bergman reproducing kernel for the space of monogenic and square integrable functions on such a domain is obtained in closed form as a finite sum of monogenic multiperiodic functions. The reproducing property leads to an estimate of the first derivative of the single‐periodic cotangent function in terms of the classical real‐valued Eisenstein series. Copyright © 2002 John Wiley Sons, Ltd.

Monogenic muitiperiodic functions in clifford analysis

In this paper we deal with generalizations of the classical Eisenstein series in the frame-work of Clifford analysis. They provide monogenic generalizations of the classical tangent, cotangent, secant and cosecant function and the elliptic functions to higher dimensions. We study properties of them and characterize them by certain duplication formulas and their principal parts.

Multiple cotangent and generalized eta functions

In this paper we shall construct multiple analogue of the cotangent functions by using the multiple Hurwitz zeta functions and study their properties and special values. In particular, we express the double cotangent functions in terms of generalized eta functions of Berndt and Lewittes.

ACOUSTIC ANALYSIS OF THE PRESSURE FIELD GENERATED BY ACCELERATION OF A TRAIN IN A LONG TUNNEL

This paper solves the acoustic wave equation for a pressure field generated by acceleration of a train in a circular tunnel of infinite length. The motion of the train is taken into account through the source term of the equation in the form of a pair of positive and negative monopoles, moving along a center axis of the tunnel, whose velocity is given by a step function of time. The solution is presented in the form of the eigenfunction-expansion and is evaluated asymptotically. The pressure field consists of the steady part due to the train moving constantly and of the transient part due to the impulsive acceleration by the step function. The former is simply the subsonic flow field around the train, while the latter is characterized by propagation of singular surfaces which are generated by the initial acceleration in the form of spherical wavefront and are reflected repeatedly at the cylindrical tunnel wall. The singularity is classified into two types, one being the type of a delta function and the other of a cotangent function. Intersections of the singular surfaces with a plane containing the axis exhibit a "diamond pattern". It is shown that this pattern disappears eventually and a distant field settles down to the one-dimensional field calculated simply by the wave equation averaged over the tunnel's cross-section.

Acoustic analysis of the pressure field in a tunnel generated by entry of a train

This paper determines the pressure field in a tunnel, generated by the entry of a travelling train. A theoretical model of the problem is first given within the framework of linear acoustic theory. The train's motion is taken into account through a source term in the wave equation, in the form of a pair of acoustic monopoles of the same magnitude but of opposite sign. An axially symmetric pressure field in a tunnel of circular cross–section and of semi–infinite length is obtained in closed form. Radiation of pressure waves into free space outside the tunnel is discarded by imposing an undisturbed boundary condition at the tunnel entrance. It is shown by asymptotic evaluation of the solutions that one–dimensional and non–dispersive wave propagation survives after the initial transients decay. This provides an estimate of the maximum pressure observed far ahead of the train. It is found from the transient solutions that two types of singularity, due to a delta function and to a cotangent function, emerge in the pressure field. The singular surfaces form a ‘diamond pattern’ in the field by repetition of reflections, at the cylindrical surface of the tunnel wall, of the spherical wavefront generated on entry into the tunnel. The magnitudes of the singularities are evaluated for the asymptotic behaviour of the pattern as time elapses. In order to check the validity of the theoretical model and analytical results, numerical computations are carried out by solving Euler equations directly, taking account of the radiation into free space. Under a situation corresponding to the theoretical model, the numerical results are found to agree well with the analytical results. As long as the blockage ratio is as small as 0.01 or 0.1, the linear acoustic theory is sufficient to describe the pressure field, even for a train Mach number 0.44. But when scattering of pressure waves by the wall edge at the tunnel entrance as the train approaches it is taken into account, the diamond pattern disappears. Instead, the transient field appears to be almost one–dimensional, which may be described by the model derived by averaging the source over the tunnel's cross–section.

Capacitance of transmission line of parallel cylinders with variable radial width

This paper presents a formulation for evaluation of the capacitance of a pair of cylinders of unequal radii with variable and different width of dielectric on the cylinders. Conformal transformation using co-tangent hyperbolic function transforms the conductor boundaries to parallel plates and dielectric boundaries to curved contours. The expression for the capacitance is derived taking into account the changes in signs of the functions representing the curved contours. Numerical data on capacitance are presented.

Impulses concealed by singularities: Transmission line theory

The input admittance of a loss-free short-circuited transmission line is generally said to be a cotangent function of frequency, a function that is imaginary and odd. The symmetry implies that the impulse response, which has the admittance function of frequency as its Fourier transform, is also odd. However, there can be no response at negative times preceding the applied impulse. Therefore the cotangent formula for susceptance cannot be correct. A real, even function of frequency representing a conductance term must be added to the imaginary, odd susceptance. Likewise, the familiar formula Z = iZ0tan(2πL/λ) is deficient.

A convenient expression for the diffraction coefficients of a wedge composed of a conductor and a lossless dielectric

The physical-optics diffraction coefficients for E-polarized diffraction by a wedge composed of a conductor and a lossless dielectric are expressed in a finite series of cotangent functions. The angular period of the cotangent functions is changed to satisfy the edge condition at the tip of the wedge, and the zeros of the cotangent functions are relocated to cancel out the incident field in the artificially complementary wedge region. It is shown that the corrected diffraction coefficients approach the exact diffraction coefficients of the perfectly conducting wedge as the relative dielectric constant increases. © 1996 John Wiley & Sons, Inc.

A convenient expression for the diffraction coefficients of a wedge composed of a conductor and a lossless dielectric

The physical-optics diffraction coefficients for E-polarized diffraction by a wedge composed of a conductor and a lossless dielectric are expressed in a finite series of cotangent functions. The angular period of the cotangent functions is changed to satisfy the edge condition at the tip of the wedge, and the zeros of the cotangent functions are relocated to cancel out the incident field in the artificially complementary wedge region. It is shown that the corrected diffraction coefficients approach the exact diffraction coefficients of the perfectly conducting wedge as the relative dielectric constant increases. © 1996 John Wiley & Sons, Inc.

A coating thickness uniformity model for physical vapour deposition systems—further validity tests

A model has been developed which considers the ratio between the coating thicknesses on sample faces towards and away from the vapour source. The model demonstrates that this ratio is an explicit hyperbolic cotangent function of ( s 2l ), where s is the source-to-substrate distance and l is a mean free path related to the distance which a vapour particle will travel before becoming thermalized. Previous work has shown the model to be valid under gas evaporation and simple diode ion-plating conditions, though investigations were generally restricted to deposition by electron beam (EB) evaporation. In this work we demonstrate the validity of the model for TiN coatings produced by reactive ion plating with discharge enhancement and for metallic coatings deposited by resistive evaporation, magnetron sputtering and cathodic arc evaporation. The results indicate how thickness uniformity can be influenced by the type of source used and we suggest that this is primarily due to the initial energy of the vapour flux and the effects of source geometry. In particular, changes in the vapour emission characteristics of EB bun sources are shown to have a significant effect.

Diffraction by an arbitrary-angled dielectric wedge. I. Physical optics approximation

A complete form is presented of the physical optics solution to diffraction by an arbitrary dielectric wedge angle with any relative dielectric constant in cases of both E- and H-polarized plane waves incident on one side of two dielectric interfaces. The solution, which is obtained by performing the physical optics (PO) approximation to the dual integral equation formulated in the spatial frequency domain, is constructed by the geometrical optics terms, including multiple reflection inside the wedge and the edge diffracted field. The diffraction coefficients of the edge diffracted field are represented in a simple form as two finite series of cotangent functions weighted by the Fresnel reflection coefficients. Far-field patterns of the PO solutions for a wedge angle of 45 degrees , relative dielectric constants 2, 10, and 100, and an E-polarized incident angle of 150 degrees are plotted in figures, revealing abrupt discontinuities at dielectric interfaces. >