On the Use of Somoff's Theorem for the Evaluation of the Elliptic Integral of the Third Species

1886 ◽  
Vol 2 (4) ◽  
pp. 73
Author(s):  
Chas H. Kummell
Keyword(s):  
Elliptic Integral ◽  
The Third

scholarly journals IX. The third elliptic integral and the ellipsotomic problem

1904 ◽  
Vol 203 (359-371) ◽  
pp. 217-304
Keyword(s):  
Elliptic Integral ◽  
Great Influence ◽  
Elliptic Functions ◽  
Mechanical Theory ◽  
The Third ◽  
History Of ◽  
Modern Analysis

The Abel Centennial Ceremony, held in Christiania, September, 1902, has directed the attention of mathematicians to the great influence of Abel on modern analysis, and. to the history of elliptic functions, and of the foundation by Crelle of the “ Journal für die reine und angewandte Mathematik.” Abel’s article in the first volume of ‘ Crelle’s Journal,' 1826, " Ueber die Integration der Differential-Formel ρdx ⁄ √R (A), wenn R und ρ gauze Functionen sind,” is of great importance as indicating the existence of what is now called the pseudo-elliptic integral; the present memoir is intended to show the utility of this integral in its application to mechanical theory.

XVIII. Researches on the geometrical properties of elliptic integrals

1852 ◽  
Vol 142 ◽  
pp. 311-416 ◽  
Keyword(s):  
Elliptic Integral ◽  
Plane Curve ◽  
Elliptic Functions ◽  
Elliptic Integrals ◽  
Mathematical Science ◽  
Third Order ◽  
Geometrical Properties ◽  
First Order ◽  
The Third ◽  
The Right

I. In placing before the Royal Society the following researches on the geometrical types of elliptic integrals, which nearly complete my investigations on this interesting subject, I may be permitted briefly to advert to what bad already been effected in this department of geometrical research. Legendre, to whom this important branch of mathematical science owes so much, devised a plane curve, whose rectification might be effected by an elliptic integral of the first order. Since that time many other geometers have followed his example, in contriving similar curves, to represent, either by their quadrature or rectification, elliptic functions. Of those who have been most successful in devising curves which should possess the required properties, may be mentioned M. Gudermann, M. Verhulst of Brussels, and M. Serret of Paris. These geometers however have succeeded in deriving from those curves scarcely any of the properties of elliptic integrals, even the most elementary. This barrenness in results was doubtless owing to the very artificial character of the genesis of those curves, devised, as they were, solely to satisfy one condition only of the general pro­blem. In 1841 a step was taken in the right direction. MM. Catalan and Gudermann, in the journals of Liouville and Crelle, showed how the arcs of spherical conic sec­tions might be represented by elliptic integrals of the third order and circular form. They did not, however, extend their investigations to the case of elliptic integrals of the third order and logarithmic form; nor even to that of the first order. These cases still remained, without any analogous geometrical representative, a blemish to the theory.

scholarly journals The third elliptic integral and the ellipsotomic problem.

1904 ◽  
Vol 73 (488-496) ◽  
pp. 1-3
Author(s):  
Alfred George Greenhill
Keyword(s):  
Numerical Calculation ◽  
Theta Function ◽  
Elliptic Integral ◽  
Exact Expression ◽  
Dynamical Problem ◽  
The Third ◽  
Circular Form

The elliptic integral of the third kind, which makes its appearance in a dynamical problem, is of the circular form in Legendre’s classification, and thus the Jacobian parameter is a fraction of the imaginary period, so that the expression by-means of the theta function can no longer be considered as reducing the variable elements from three to two. Burkhardt has given a series rapidly convergent for the numerical calculation of such cases; but the object of this memoir is to develop the exact expression by means of an idea of Abel, given in the first volume of ‘Crelle’s Journal,’ 1826, “Ueber die Integration der Differential-Formel (1) ρdx /√R, wenn R and ρ ganze Functionen sind."

Interaction Behaviours Between Soliton and Cnoidal Periodic Waves for the Cubic Generalised Kadomtsev–Petviashvili Equation

2015 ◽  
Vol 70 (7) ◽  
pp. 539-544 ◽  
Author(s):  
Bo Ren ◽  
Ji Lin
Keyword(s):  
Elliptic Integral ◽  
Elliptic Functions ◽  
Jacobi Elliptic Functions ◽  
Periodic Waves ◽  
The Third ◽  
Interaction Solutions ◽  
Petviashvili Equation ◽  
Kink Soliton

AbstractThe consistent tanh expansion (CTE) method is applied to the cubic generalised Kadomtsev–Petviashvili (CGKP) equation. The interaction solutions between one kink soliton and the cnoidal periodic waves are explicitly given. Some special concrete interaction solutions in terms of the Jacobi elliptic functions and the third type of incomplete elliptic integral are discussed both in analytical and graphical ways.

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