The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform
Jeanmarc Hure - One of the best experts on this subject based on the ideXlab platform.
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a new equation for the mid plane potential of power law discs ii exact solutions and approximate formulae
Astronomy and Astrophysics, 2008Co-Authors: Jeanmarc Hure, F Hersant, C Carreau, J P BussetAbstract:Aims. The first-order ordinary differential equation (ODE) that describes the mid-plane gravitational potential in flat finite size discs of surface density Σ(R) ∝ R s (Hure & Hersant 2007, A&A, 467, 907) is solved exactly in terms of infinite series. Methods. The formal solution of the ODE is derived and then converted into a series representation by expanding the Elliptic Integral of the first kind over its modulus before analytical integration. Results. Inside the disc, the gravitational potential consists of three terms: a power law of radius R with index 1 + s ,a nd two infinite series of the variables R and 1/R. The convergence of the series can be accelerated, enabling the construction of reliable approximations. At the lowest-order, the potential inside large astrophysical discs (s ∼− 1.5 ± 1) is described by a very simple formula whose accuracy (a few percent typically) is easily increased by considering successive orders through a recurrence. A basic algorithm is given. Conclusions. Applications concern all theoretical models and numerical simulations where the influence of disc gravity must be checked and/or reliably taken into account.
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a new equation for the mid plane potential of power law discs ii exact solutions and approximate formulae
arXiv: Astrophysics, 2008Co-Authors: Jeanmarc Hure, F Hersant, C Carreau, J P BussetAbstract:The first-order ordinary differential equation (ODE) that describes the mid-plane gravitational potential in flat finite size discs in which the surface density is a power-law function of the radius R with exponent s (Hur\'e & Hersant 2007) is solved exactly in terms of infinite series. The formal solution of the ODE is derived and then converted into a series representation by expanding the Elliptic Integral of the first kind over its modulus before analytical integration. Inside the disc, the gravitational potential consists of three terms: a power law of radius R with index 1+s, and two infinite series of the variables R and 1/R. The convergence of the series can be accelerated, enabling the construction of reliable approximations. At the lowest-order, the potential inside large astrophysical discs (s ~ -1.5 +/- 1) is described by a very simple formula whose accuracy (a few percent typically) is easily increased by considering successive orders through a recurrence. A basic algorithm is given. Applications concern all theoretical models and numerical simulations where the influence of disc gravity must be checked and/or reliably taken into account.
J P Busset - One of the best experts on this subject based on the ideXlab platform.
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a new equation for the mid plane potential of power law discs ii exact solutions and approximate formulae
Astronomy and Astrophysics, 2008Co-Authors: Jeanmarc Hure, F Hersant, C Carreau, J P BussetAbstract:Aims. The first-order ordinary differential equation (ODE) that describes the mid-plane gravitational potential in flat finite size discs of surface density Σ(R) ∝ R s (Hure & Hersant 2007, A&A, 467, 907) is solved exactly in terms of infinite series. Methods. The formal solution of the ODE is derived and then converted into a series representation by expanding the Elliptic Integral of the first kind over its modulus before analytical integration. Results. Inside the disc, the gravitational potential consists of three terms: a power law of radius R with index 1 + s ,a nd two infinite series of the variables R and 1/R. The convergence of the series can be accelerated, enabling the construction of reliable approximations. At the lowest-order, the potential inside large astrophysical discs (s ∼− 1.5 ± 1) is described by a very simple formula whose accuracy (a few percent typically) is easily increased by considering successive orders through a recurrence. A basic algorithm is given. Conclusions. Applications concern all theoretical models and numerical simulations where the influence of disc gravity must be checked and/or reliably taken into account.
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a new equation for the mid plane potential of power law discs ii exact solutions and approximate formulae
arXiv: Astrophysics, 2008Co-Authors: Jeanmarc Hure, F Hersant, C Carreau, J P BussetAbstract:The first-order ordinary differential equation (ODE) that describes the mid-plane gravitational potential in flat finite size discs in which the surface density is a power-law function of the radius R with exponent s (Hur\'e & Hersant 2007) is solved exactly in terms of infinite series. The formal solution of the ODE is derived and then converted into a series representation by expanding the Elliptic Integral of the first kind over its modulus before analytical integration. Inside the disc, the gravitational potential consists of three terms: a power law of radius R with index 1+s, and two infinite series of the variables R and 1/R. The convergence of the series can be accelerated, enabling the construction of reliable approximations. At the lowest-order, the potential inside large astrophysical discs (s ~ -1.5 +/- 1) is described by a very simple formula whose accuracy (a few percent typically) is easily increased by considering successive orders through a recurrence. A basic algorithm is given. Applications concern all theoretical models and numerical simulations where the influence of disc gravity must be checked and/or reliably taken into account.
Yuming Chu - One of the best experts on this subject based on the ideXlab platform.
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approximations for the complete Elliptic Integral of the second hbox kind kind
Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie A-matematicas, 2021Co-Authors: Weimao Qian, Miao-kun Wang, Yuming ChuAbstract:In this paper, we present the best possible parameters $$\alpha _{1}$$ , $$\alpha _{2}$$ , $$\alpha _{3}$$ , $$\alpha _{4}$$ , $$\beta _{1}$$ , $$\beta _{2}$$ , $$\beta _{3}$$ , $$\beta _{4} \in {\mathbb {R}}$$ such that $$\begin{aligned} \frac{\alpha _{1}}{H(x, y)}+\frac{1-\alpha _{1}}{L(x, y)}&<\frac{1}{V(x, y)}<\frac{\beta _{1}}{H(x, y)}+\frac{1-\beta _{1}}{L(x, y)},\\ \frac{\alpha _{2}}{H(x, y)}+\frac{1-\alpha _{2}}{P(x, y)}&<\frac{1}{V(x, y)}<\frac{\beta _{2}}{H(x, y)}+\frac{1-\beta _{2}}{P(x, y)},\\ \frac{\alpha _{3}}{H(x, y)}+\frac{1-\alpha _{3}}{N S(x, y)}&<\frac{1}{V(x, y)}<\frac{\beta _{3}}{H(x, y)}+\frac{1-\beta _{3}}{N S(x, y)},\\ \frac{\alpha _{4}}{H(x, y)}+\frac{1-\alpha _{4}}{T(x, y)}&<\frac{1}{V(x, y)}<\frac{\beta _{4}}{H(x, y)}+\frac{1-\beta _{4}}{T(x, y)} \end{aligned}$$ hold for all $$x, y>0$$ with $$x \ne y$$ , where H(x, y), G(x, y), L(x, y), A(x, y), NS(x, y), P(x, y) and T(x, y) are respectively the harmonic, geometric, logarithmic, arithmetic, Neuman-Sandor, and first and second Seiffert means of two distinct positive numbers x and y, and $$\begin{aligned} V(x,y)=\pi G^{2}(x, y) /\left[ 2\int _{0}^{\pi /2}\sqrt{A^{2}(x,y) \cos ^{2}\varphi +G^{2}(x,y)\sin ^{2}\varphi }d\varphi \right] \end{aligned}$$ is a new Seiffert-like mean. As applications, some new inequalities for the complete Elliptic Integral of the second kind are given.
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monotonicity and convexity involving generalized Elliptic Integral of the first kind
Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie A-matematicas, 2021Co-Authors: Tiehong Zhao, Miao-kun Wang, Yuming ChuAbstract:In this paper, we present the monotonicity properties of the ratio between generalized Elliptic Integral of the first kind $${\mathcal {K}}_a(r)$$ and its approximation $$\log [1+2/(ar')]$$ , and also the convexity (concavity) of their difference for $$a\in (0,1/2]$$ . As an application, we give new bounds for generalized Grotzsch ring function $$\mu _a(r)$$ and a upper bound for $${\mathcal {K}}_a(r)$$ .
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a sharp double inequality involving generalized complete Elliptic Integral of the first kind
Math 2020 Vol. 5 Pages 4512-4528, 2020Co-Authors: Tiehong Zhao, Miao-kun Wang, Yuming ChuAbstract:In the article, we establish a sharp double inequality involving the ratio of generalized complete Elliptic Integrals of the first kind, which is the improvement and generalization of some previously known results.
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approximation for the complete Elliptic Integral of the first kind
Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie A-matematicas, 2020Co-Authors: Weimao Qian, Yuming ChuAbstract:In the article, we present several sharp upper and lower bounds for the complete Elliptic Integral of the first kind in terms of inverse trigonometric and inverse hyperbolic functions. As consequences, some sharp bounds for the Gaussian arithmetic-geometric mean in terms of other bivariate means are also given.
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sharp power mean inequalities for the generalized Elliptic Integral of the first kind
Computational Methods and Function Theory, 2020Co-Authors: Miao-kun Wang, Yuming ChuAbstract:We establish two sharp inequalities involving the power mean and generalized Elliptic Integral of the first kind. As applications, the analogous inequalities concerning the complete p-Elliptic Integral of the first kind are also derived.
Miao-kun Wang - One of the best experts on this subject based on the ideXlab platform.
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approximations for the complete Elliptic Integral of the second hbox kind kind
Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie A-matematicas, 2021Co-Authors: Weimao Qian, Miao-kun Wang, Yuming ChuAbstract:In this paper, we present the best possible parameters $$\alpha _{1}$$ , $$\alpha _{2}$$ , $$\alpha _{3}$$ , $$\alpha _{4}$$ , $$\beta _{1}$$ , $$\beta _{2}$$ , $$\beta _{3}$$ , $$\beta _{4} \in {\mathbb {R}}$$ such that $$\begin{aligned} \frac{\alpha _{1}}{H(x, y)}+\frac{1-\alpha _{1}}{L(x, y)}&<\frac{1}{V(x, y)}<\frac{\beta _{1}}{H(x, y)}+\frac{1-\beta _{1}}{L(x, y)},\\ \frac{\alpha _{2}}{H(x, y)}+\frac{1-\alpha _{2}}{P(x, y)}&<\frac{1}{V(x, y)}<\frac{\beta _{2}}{H(x, y)}+\frac{1-\beta _{2}}{P(x, y)},\\ \frac{\alpha _{3}}{H(x, y)}+\frac{1-\alpha _{3}}{N S(x, y)}&<\frac{1}{V(x, y)}<\frac{\beta _{3}}{H(x, y)}+\frac{1-\beta _{3}}{N S(x, y)},\\ \frac{\alpha _{4}}{H(x, y)}+\frac{1-\alpha _{4}}{T(x, y)}&<\frac{1}{V(x, y)}<\frac{\beta _{4}}{H(x, y)}+\frac{1-\beta _{4}}{T(x, y)} \end{aligned}$$ hold for all $$x, y>0$$ with $$x \ne y$$ , where H(x, y), G(x, y), L(x, y), A(x, y), NS(x, y), P(x, y) and T(x, y) are respectively the harmonic, geometric, logarithmic, arithmetic, Neuman-Sandor, and first and second Seiffert means of two distinct positive numbers x and y, and $$\begin{aligned} V(x,y)=\pi G^{2}(x, y) /\left[ 2\int _{0}^{\pi /2}\sqrt{A^{2}(x,y) \cos ^{2}\varphi +G^{2}(x,y)\sin ^{2}\varphi }d\varphi \right] \end{aligned}$$ is a new Seiffert-like mean. As applications, some new inequalities for the complete Elliptic Integral of the second kind are given.
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monotonicity and convexity involving generalized Elliptic Integral of the first kind
Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie A-matematicas, 2021Co-Authors: Tiehong Zhao, Miao-kun Wang, Yuming ChuAbstract:In this paper, we present the monotonicity properties of the ratio between generalized Elliptic Integral of the first kind $${\mathcal {K}}_a(r)$$ and its approximation $$\log [1+2/(ar')]$$ , and also the convexity (concavity) of their difference for $$a\in (0,1/2]$$ . As an application, we give new bounds for generalized Grotzsch ring function $$\mu _a(r)$$ and a upper bound for $${\mathcal {K}}_a(r)$$ .
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a sharp double inequality involving generalized complete Elliptic Integral of the first kind
Math 2020 Vol. 5 Pages 4512-4528, 2020Co-Authors: Tiehong Zhao, Miao-kun Wang, Yuming ChuAbstract:In the article, we establish a sharp double inequality involving the ratio of generalized complete Elliptic Integrals of the first kind, which is the improvement and generalization of some previously known results.
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sharp power mean inequalities for the generalized Elliptic Integral of the first kind
Computational Methods and Function Theory, 2020Co-Authors: Miao-kun Wang, Yuming ChuAbstract:We establish two sharp inequalities involving the power mean and generalized Elliptic Integral of the first kind. As applications, the analogous inequalities concerning the complete p-Elliptic Integral of the first kind are also derived.
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optimal combinations bounds of root square and arithmetic means for toader mean
Proceedings - Mathematical Sciences, 2012Co-Authors: Miao-kun WangAbstract:We find the greatest value α1 and α2, and the least values β1 and β2, such that the double inequalities α1S(a,b) + (1 − α1) A(a,b) 0 with a ≠ b. As applications, we get two new bounds for the complete Elliptic Integral of the second kind in terms of elementary functions. Here, S(a,b) = [(a2 + b2)/2]1/2, A(a,b) = (a + b)/2, and \(T(a,b)=\frac{2}{\pi}\int\limits_{0}^{{\pi}/{2}}\sqrt{a^2{\cos^2{\theta}}+b^2{\sin^2{\theta}}}{\rm d}\theta\) denote the root-square, arithmetic, and Toader means of two positive numbers a and b, respectively.
C Carreau - One of the best experts on this subject based on the ideXlab platform.
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a new equation for the mid plane potential of power law discs ii exact solutions and approximate formulae
Astronomy and Astrophysics, 2008Co-Authors: Jeanmarc Hure, F Hersant, C Carreau, J P BussetAbstract:Aims. The first-order ordinary differential equation (ODE) that describes the mid-plane gravitational potential in flat finite size discs of surface density Σ(R) ∝ R s (Hure & Hersant 2007, A&A, 467, 907) is solved exactly in terms of infinite series. Methods. The formal solution of the ODE is derived and then converted into a series representation by expanding the Elliptic Integral of the first kind over its modulus before analytical integration. Results. Inside the disc, the gravitational potential consists of three terms: a power law of radius R with index 1 + s ,a nd two infinite series of the variables R and 1/R. The convergence of the series can be accelerated, enabling the construction of reliable approximations. At the lowest-order, the potential inside large astrophysical discs (s ∼− 1.5 ± 1) is described by a very simple formula whose accuracy (a few percent typically) is easily increased by considering successive orders through a recurrence. A basic algorithm is given. Conclusions. Applications concern all theoretical models and numerical simulations where the influence of disc gravity must be checked and/or reliably taken into account.
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a new equation for the mid plane potential of power law discs ii exact solutions and approximate formulae
arXiv: Astrophysics, 2008Co-Authors: Jeanmarc Hure, F Hersant, C Carreau, J P BussetAbstract:The first-order ordinary differential equation (ODE) that describes the mid-plane gravitational potential in flat finite size discs in which the surface density is a power-law function of the radius R with exponent s (Hur\'e & Hersant 2007) is solved exactly in terms of infinite series. The formal solution of the ODE is derived and then converted into a series representation by expanding the Elliptic Integral of the first kind over its modulus before analytical integration. Inside the disc, the gravitational potential consists of three terms: a power law of radius R with index 1+s, and two infinite series of the variables R and 1/R. The convergence of the series can be accelerated, enabling the construction of reliable approximations. At the lowest-order, the potential inside large astrophysical discs (s ~ -1.5 +/- 1) is described by a very simple formula whose accuracy (a few percent typically) is easily increased by considering successive orders through a recurrence. A basic algorithm is given. Applications concern all theoretical models and numerical simulations where the influence of disc gravity must be checked and/or reliably taken into account.