scholarly journals On the theory of the elliptic transcendents

1837 ◽  
Vol 3 ◽  
pp. 60-60
Keyword(s):  
Elliptic Function ◽  
Elliptic Integral ◽  
Elliptic Functions ◽  
Algebraic Expression ◽  
Algebraic Equations ◽  
Similar Function ◽  
Circular Arcs ◽  
Straight Line ◽  
Regular Theory ◽  
General Method

Fagnani discovered that the two arcs of the periphery of a given ellipse may be determined in many ways, so that their difference shall be equal to an assignable straight line; and proved that any arc of a lemniscate, like that of a circle, may be multiplified any number of times, or may be subdivided into any number of equal parts, by finite algebraic equations. What he had accomplished with respect to the arcs of the lemniscates, which are expressed by a particular elliptic integral, Euler extended to all transcendents of the same class. Landen showed that the arcs of the hyperbola may be reduced, by a proper transformation, to those of an ellipse. Lagrange furnished us with a general method for changing an elliptic function into another having a different modulus; a process which greatly facilitates the numerical calculation of this class of integrals. Legendre distributed the elliptic functions into distinct classes, and reduced them to a regular theory, developing many of their properties which were before unknown, and introducing many important additions and improvements in the theory. Mr. Abel of Christiana happity conceived the idea of expressing the amplitude of an elliptic function in terms of the function itself, which led to the discovery of many new and useful properties. Mr. Jacobi proved, by a different method, that an elliptic function may be transformed in innumerable ways into another similar function, to which it bears constantly the same proportion. But his demonstrations require long and complicated calculations; and the train of deductions he pursues does not lead naturally to the truths which are proved, nor does it present in a connected view all the conclusions which the theory embraces. The author of the present paper gives a comprehensive view of the theory in its full extent, and deduces all the connected truths from the same principle. He finds that the sines or cosines of the amplitudes, used in the transformations, are analogous to the sines or cosines of two circular arcs, one of which is a multiple of the other; so that the former quantities are changed into the latter when the modulus is supposed to vanish in the algebraic expression. Hence he is enabled to transfer to the elliptic transcendents the same methods of investigation that succeed in the circle: a procedure which renders the demonstrations considerably shorter, and which removes most of the difficulties, in consequence of the close analogy that subsists between the two cases.

scholarly journals XX. On the theory of the elliptic transcendents

1831 ◽  
Vol 121 ◽  
pp. 349-377 ◽  
Keyword(s):  
Algebraic Equation ◽  
Elliptic Function ◽  
Elliptic Integral ◽  
Elliptic Functions ◽  
The Other ◽  
Algebraic Equations ◽  
Integral Calculus ◽  
Circular Arc ◽  
Regular Theory ◽  
The Difference

The branch of the integral calculus which treats of elliptic transcendents originated in the researches of Fagnani, an Italian geometer of eminence. He discovered that two arcs of the periphery of a given ellipse may be determined in many ways, so that their difference shall be equal to an assignable straight line; and he proved that any arc of the lemniscata, like that of a circle, may be multiplied any number of times, or may be subdivided into any number of equal parts, by finite algebraic equations. These are particular results; and it was the discoveries of Euler that enabled geometers to advance to the investigation of the general properties of the elliptic functions. An integral in finite terms deduced by that geometer from an equation between the differentials of two similar transcendent quantities not separately integrable, led immediately to an algebraic equation between the amplitudes of three elliptic functions, of which one is the sum, or the difference, of the other two. This sort of integrals, therefore, could now be added or subtracted in a manner analogous to circular arcs, or logarithms; the amplitude of the sum, or of the difference, being expressed algebraically by means of the amplitudes of the quantities added or subtracted. What Fagnani had accomplished with respect to the arcs of the lemniscata, which are expressed by a particular elliptic integral, Euler extended to all transcendents of the same class. To multiply a function of this kind, or to subdivide it into equal parts, was reduced to solving an algebraic equation. In general, all the properties of the elliptic transcendents, in which the modulus remains unchanged, are deducible from the discoveries of Euler. Landen enlarged our knowledge of this kind of functions, and made a useful addition to analysis, by showing that the arcs of the hyperbola may be reduced, by a proper transformation, to those of the ellipse. Every part of analysis is indebted to Lagrange, who enriched this particular branch with a general method for changing an elliptic function into another having a different modulus, a process which greatly facilitates the numerical calculation of this class of integrals. An elliptic function lies between an arc of the circle on one hand, and a logarithm on the other, approaching indefinitely to the first when the modulus is diminished to zero, and to the second when the modulus is augmented to unit, its other limit. By repeatedly applying the transformation of Lagrange, we may compute either a scale of decreasing moduli reducing the integral to a circular arc, or a scale of increasing moduli bringing it continually nearer to a logarithm. The approximation is very elegant and simple, and attains the end proposed with great rapidity. The discoveries that have been mentioned occurred in the general cultivation of analysis; but Legendre has bestowed much of his attention and study upon this particular branch of the integral calculus. He distributed the elliptic functions in distinct classes, and reduced them to a regular theory. In a Mémoire sur les Transcendantes Elliptiques, published in 1793, and in his Exercices de Calcul Intégral, which appeared in 1817 he has developed many of their properties entirely new; investigated the easiest methods of approximating to their values; computed numerical tables to facilitate their application; and exemplified their use in some interesting problems of geometry and mechanics. In a publication so late as 1825, the author, returning to the same subject, has rendered his theory still more perfect, and made many additions to it which further researches had suggested. In particular we find a new method of making an elliptic function approach as near as we please to a circular arc, or to a logarithm, by a scale of reduction very different from that of which Lagrange is the author, the only one before known. This step in advance would unavoidably have conducted to a more extensive theory of this kind of integrals, which, nearly about the same time, was being discovered by the researches of other geometers.

scholarly journals Frequency–Amplitude Relationship of a Nonlinear Symmetric Panel Absorber Mounted on a Flexible Wall

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1188
Author(s):  
Yiu-Yin Lee
Keyword(s):  
Elliptic Integral ◽  
Algebraic Equations ◽  
Cavity Depth ◽  
Nonlinear Algebraic Equations ◽  
Flexible Panel ◽  
Flexible Wall ◽  
Elliptic Integral Method ◽  
Flexible Panels ◽  
Relationship Of ◽  
Panel Absorber

This study addresses the frequency–amplitude relationship of a nonlinear symmetric panel absorber mounted on a flexible wall. In many structural–acoustic works, only one flexible panel is considered in their models with symmetric configuration. There are very limited research investigations that focus on two flexible panels coupled with a cavity, particularly for nonlinear structural–acoustic problems. In practice, panel absorbers with symmetric configurations are common and usually mounted on a flexible wall. Thus, it should not be assumed that the wall is rigid. This study is the first work employing the weighted residual elliptic integral method for solving this problem, which involves the nonlinear multi-mode governing equations of two flexible panels coupled with a cavity. The reason for adopting the proposed solution method is that fewer nonlinear algebraic equations are generated. The results obtained from the proposed method and finite element method agree reasonably well with each other. The effects of some parameters such as vibration amplitude, cavity depth and thickness ratio, etc. are also investigated.

ANALYTICAL STUDY ON NONLINEAR DIFFERENTIAL–DIFFERENCE EQUATIONS VIA A NEW METHOD

2010 ◽  
Vol 24 (08) ◽  
pp. 761-773
Author(s):  
HONG ZHAO
Keyword(s):  
Difference Equations ◽  
Elliptic Function ◽  
Analytical Study ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Periodic Wave ◽  
Jacobian Elliptic Function ◽  
Jacobian Elliptic Functions ◽  
Periodic Wave Solutions ◽  
Trigonometric Function Solutions

Based on the computerized symbolic computation, a new rational expansion method using the Jacobian elliptic function was presented by means of a new general ansätz and the relations among the Jacobian elliptic functions. The results demonstrated an effective direction in terms of a uniformed construction of the new exact periodic solutions for nonlinear differential–difference equations, where two representative examples were chosen to illustrate the applications. Various periodic wave solutions, including Jacobian elliptic sine function, Jacobian elliptic cosine function and the third elliptic function solutions, were obtained. Furthermore, the solitonic solutions and trigonometric function solutions were also obtained within the limit conditions in this paper.

A General Method for the Fatigue-Resistant Design of Mechanical Components—Part 2: Analytical

1975 ◽  
Vol 97 (3) ◽  
pp. 970-975
Author(s):  
D. T. Vaughan ◽  
L. D. Mitchell
Keyword(s):  
Analytical Solution ◽  
Fatigue Fracture ◽  
Fracture Criterion ◽  
Fatigue Loading ◽  
General Analytical Solution ◽  
Failure Theory ◽  
Straight Line ◽  
Mechanical Components ◽  
General Method

This paper develops the general analytical solution to the design of mechanical components under fatigue loading. Its only limitation is that the overloading lines must be a straight line on the σa−σm diagram. The designer is free to select his own failure theory for the material he intends to use as well as to select his own fatigue fracture criterion.

Analytical Solutions of Two Space-Time Fractional Nonlinear Models Using Jacobi Elliptic Function Expansion Method

2021 ◽  
pp. 173-188
Author(s):  
Zillur Rahman ◽  
M. Zulfikar Ali ◽  
Harun-Or-Roshid ◽  
Mohammad Safi Ullah
Keyword(s):  
Elliptic Function ◽  
Nonlinear Models ◽  
Applied Mathematics ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Space Time ◽  
Jacobi Elliptic Function ◽  
Function Expansion ◽  
Partial Models ◽  
Jacobi Elliptic Function Expansion

In this manuscript, the space-time fractional Equal-width (s-tfEW) and the space-time fractional Wazwaz-Benjamin-Bona-Mahony (s-tfWBBM) models have been investigated which are frequently arises in nonlinear optics, solid states, fluid mechanics and shallow water. Jacobi elliptic function expansion integral technique has been used to build more innovative exact solutions of the s-tfEW and s-tfWBBM nonlinear partial models. In this research, fractional beta-derivatives are applied to convert the partial models to ordinary models. Several types of solutions have been derived for the models and performed some new solitary wave phenomena. The derived solutions have been presented in the form of Jacobi elliptic functions initially. Persevering different conditions on a parameter, we have achieved hyperbolic and trigonometric functions solutions from the Jacobi elliptic function solutions. Besides the scientific derivation of the analytical findings, the results have been illustrated graphically for clear identification of the dynamical properties. It is noticeable that the integral scheme is simplest, conventional and convenient in handling many nonlinear models arising in applied mathematics and the applied physics to derive diverse structural precise solutions.

Appendix, containing the Investigation of a Formula for the Rectification of any Arch of an Ellipse

1805 ◽  
Vol 5 (2) ◽  
pp. 271-293
Keyword(s):  
Finite Number ◽  
The Other ◽  
Algebraic Equations ◽  
Conic Sections ◽  
Straight Line ◽  
Curve Line ◽  
Straight Lines

It is now generally understood, that by the rectification of a curve line, is meant, not only the method of finding a straight line exactly equal to it, but also the method of expressing it by certain functions of the other lines, whether straight lines or circles, by which the nature of the curve is defined. It is evidently in the latter sense that we must understand the term rectification, when applied to the arches of conic sections, seeing that it has hitherto been found impossible, either to exhibit straight lines equal to them, or to express their relation to their co-ordinates, by algebraic equations, consisting of a finite number of terms.

Grouping of Straight Line Segments and Circular Arcs for Scene Analysis

1995 ◽  
pp. 163-185
Author(s):  
Jean-Luc Arseneault ◽  
Robert Bergevin ◽  
Denis Laurendeau
Keyword(s):  
Scene Analysis ◽  
Line Segments ◽  
Circular Arcs ◽  
Straight Line

scholarly journals IV. A general method for exhibiting the value of an algebraic expression involving several radical quantities in an infinite series: wherein Sir Isaac Newton’s theorem for involving a binomial, with another of the same author, relating to the roots of equations, are demonstrated

1752 ◽  
Vol 47 ◽  
pp. 20-27 ◽  
Keyword(s):  
Infinite Series ◽  
Algebraic Expression ◽  
Roots Of Equations ◽  
General Method

Among all the great improvements, which the art of computation hath in these last ages received, the most considerable; since not only the doctrine of chances and annuites, with some other branches of the mathematics, depend almost intirely thereon, but even the business of fluents, of such extensive use, would, without its aid and concurrence, be quite at a stand in a multitude of cases, as is well known to mathematicians.

Numerical approximations for fractional diffusion equations via a Chebyshev spectral-tau method

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Eid Doha ◽  
Ali Bhrawy ◽  
Samer Ezz-Eldien
Keyword(s):  
Numerical Technique ◽  
Operational Matrix ◽  
Fractional Diffusion ◽  
High Accuracy ◽  
Variable Coefficients ◽  
Diffusion Equations ◽  
Algebraic Equations ◽  
Fractional Differentiation ◽  
Fractional Diffusion Equations ◽  
General Method

AbstractIn this paper, a class of fractional diffusion equations with variable coefficients is considered. An accurate and efficient spectral tau technique for solving the fractional diffusion equations numerically is proposed. This method is based upon Chebyshev tau approximation together with Chebyshev operational matrix of Caputo fractional differentiation. Such approach has the advantage of reducing the problem to the solution of a system of algebraic equations, which may then be solved by any standard numerical technique. We apply this general method to solve four specific examples. In each of the examples considered, the numerical results show that the proposed method is of high accuracy and is efficient for solving the time-dependent fractional diffusion equations.

Hausdorff Dimension of Sets of Escaping Points and Escaping Parameters for Elliptic Functions

2015 ◽  
Vol 59 (3) ◽  
pp. 671-690
Author(s):  
Piotr Gałązka ◽  
Janina Kotus
Keyword(s):  
Hausdorff Dimension ◽  
Elliptic Function ◽  
Meromorphic Functions ◽  
Elliptic Functions ◽  
Critical Value ◽  
Parameter Plane ◽  
Dynamical Plane ◽  
The Family ◽  
Maximal Multiplicity ◽  
Additional Assumptions

AbstractLetbe a non-constant elliptic function. We prove that the Hausdorff dimension of the escaping set offequals 2q/(q+1), whereqis the maximal multiplicity of poles off. We also consider theescaping parametersin the familyfβ=βf, i.e. the parametersβfor which the orbit of one critical value offβescapes to infinity. Under additional assumptions onfwe prove that the Hausdorff dimension of the set of escaping parametersεin the familyfβis greater than or equal to the Hausdorff dimension of the escaping set in the dynamical space. This demonstrates an analogy between the dynamical plane and the parameter plane in the class of transcendental meromorphic functions.

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