A Rational Approximation for the Complete Elliptic Integral of the First Kind [PDF]
Mathematics, 2020 Let K ( r ) be the complete elliptic integral of the first kind. We present an accurate rational lower approximation for K ( r ) . More precisely, we establish the inequality 2 π K ( r ) > 5 ( r ′ ) 2 + 126 r ′ + 61 61 (
Zhen-Hang Yang +2 more
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Foliations in algebraic surfaces having a rational first integral [PDF]
Publicacions Matemàtiques, 1997 Given a foliation F in an algebraic surface having a rational first integral a genus formula for the general solution is obtained. In the case S = P2 some new counter-examples to the classic formulation of the Poincar'e problem are presented.
García Zamora, Alexis
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Darboux and rational first integrals for a family of cubic three dimensional system
Zanco Journal of Pure and Applied Sciences, 2021 In this paper, we investigate the first integrals of the following system where and , . This kind of system is a special case of three-dimensional polynomial cubic differential systems.
Sarbast H. Mikaeel , Azad I. Amen
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Rational first integrals of the Liénard equations: The solution to the Poincaré problem for the Liénard equations [PDF]
Anais da Academia Brasileira de Ciências, 2021 Poincaré in 1891 asked about the necessary and sufficient conditions in order to characterize when a polynomial differential system in the plane has a rational first integral. Here we solve this question for the class of Liénard differential equations
JAUME LLIBRE +2 more
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Finding Liouvillian first integrals of rational ODEs of any order in finite terms [PDF]
Symmetry, Integrability and Geometry: Methods and Applications, 2005 It is known, due to Mordukhai-Boltovski, Ritt, Prelle, Singer, Christopher and others, that if a given rational ODE has a Liouvillian first integral then the corresponding integrating factor of the ODE must be of a very special form of a product of ...
Kosovtsov, Yuri N.
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Rational first integrals in the Darboux theory of integrability in Cn [PDF]
International Journal of Bifurcation and Chaos, 2008 In 1976 Jouanolou showed that if the number of invariant algebraic hypersurfaces of a polynomial vector field in Rn or Cn of degree d is at least (d + n/n- 1) + n, then the vector field has a rational first integral. His proof used sophisticated tools of algebraic geometry.
Llibre, Jaume +2 more
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Polynomial and rational first integrals for planar homogeneous polynomial differential systems [PDF]
Publicacions Matemàtiques, 2021 In this paper we find necessary and suficient conditions in order that a planar homogeneous polynomial differential system has a polynomial or rational first integral. We apply these conditions to linear and quadratic homogeneous polynomial differential systems.
Giné, Jaume, Grau, Maite, Llibre, Jaume
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Darboux and rational first integrals for a family of cubic three dimensional systems
Zanco Journal of Pure and Applied Sciences, 2021 In this paper, we investigate the first integrals of the following system exam.PNG exam1.PNG This kind of system is a particular case of the jerk cubic three dimensional polynomial differential systems.
Sarbast Hussein Mikaeel, Azad I. Amen
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Interpolation rational integral fraction of the Hermitian-type on a continual set of nodes
Математичні Студії, 2021 Some approaches to the construction of interpolation rational integral approximations with arbitrary multiplicity of nodes are analyzed. An integral rational Hermitian-type interpolant of the third order on a continual set of nodes, which is the ratio of
Ya. O. Baranetskij +3 more
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New Lower Bound for the Generalized Elliptic Integral of the First Kind
Mathematics, 2022 In this paper, we obtain a new simple rational approximation for Ka(r): the inequality 2Ka(r)/π>g2r′/g1r′ holds for all r∈(0,1), where Ka(r) is the generalized elliptic integral of the first kind, r′=1−r2, g1r′ and g2r′ are specific primary and quadratic
Ling Zhu
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