List questions from elliptic-integrals

Published on 14 Jul 2020 - 17:45

149

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How does $\int_0^{2\pi}\sqrt{(a-\cos\theta)^2+\sin^2\theta}\ d\theta$ end up as an elliptic integral of the second kind?

Published on 19 Jul 2020 - 7:41

106

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Find the constant for $\int_{0}^{1} {\frac{\mathrm{d}x}{\sqrt{(1-x^2)(1-(kx)^4)}}} \sim C\ln(1-k)$

Published on 22 Jul 2020 - 3:07

71

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Integrals of the form $\int_a^b\frac1{F(x)}\sin(\int_a^x F(u) du) dx$, in particular with $F(x)=\sqrt{p_4(x)}h(x)$, for $p_4$ a quartic polynomial

Published on 22 Jul 2020 - 14:13

953

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Solving what Mathematica could not

Published on 11 Jul 2016 - 18:19

166

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Asymptotic expansion of elliptic integral

Published on 18 Jul 2016 - 9:32

489

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To find area of the curves that are extension of ellipse

Published on 02 Aug 2016 - 14:01

277

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Closed form for the integral $\int_0^\infty \frac{dx}{\sqrt{(x+a)(x+b)(x+c)(x+d)}}$

Published on 04 Aug 2016 - 12:11

331

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To find solution of $y''(x)=ay^3(x)+by^2(x)+cy(x)+d$

Published on 05 Aug 2016 - 8:02

121

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How to prove Maxwell's mutual inductance between two parallel circular loops integral?

Published on 10 Aug 2016 - 20:56

171

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Prove that this family of iterated means gives odd powers of Arithmetic-Geometric Mean in the limit

Published on 20 Aug 2016 - 0:49

326

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Prove $\int_0^\infty \frac{dx}{\sqrt{(x^4+a^4)(x^4+b^4)}}=\frac{\pi}{2 \sqrt2 a b} ( \text{agm} (\frac{a+b}{2},\sqrt{\frac{a^2+b^2}{2}} ))^{-1}$

Published on 27 Aug 2016 - 15:20

644

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Calculate $\int_0^1\frac{\cos(\arctan x)}{\sqrt{x}}dx$

Published on 15 Sep 2016 - 16:35

393

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Principal value of complete elliptic integral of third kind

Published on 25 Mar 2026 - 14:34

218

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Principal value of Carlson elliptic integral $R_C$

Published on 25 Mar 2026 - 14:33

List Question

Solution to a Nonlinear System of ODEs

How does $\int_0^{2\pi}\sqrt{(a-\cos\theta)^2+\sin^2\theta}\ d\theta$ end up as an elliptic integral of the second kind?

Find the constant for $\int_{0}^{1} {\frac{\mathrm{d}x}{\sqrt{(1-x^2)(1-(kx)^4)}}} \sim C\ln(1-k)$

Integrals of the form $\int_a^b\frac1{F(x)}\sin(\int_a^x F(u) du) dx$, in particular with $F(x)=\sqrt{p_4(x)}h(x)$, for $p_4$ a quartic polynomial

Solving what Mathematica could not

Asymptotic expansion of elliptic integral

To find area of the curves that are extension of ellipse

Closed form for the integral $\int_0^\infty \frac{dx}{\sqrt{(x+a)(x+b)(x+c)(x+d)}}$

To find solution of $y''(x)=ay^3(x)+by^2(x)+cy(x)+d$

How to prove Maxwell's mutual inductance between two parallel circular loops integral?

Prove that this family of iterated means gives odd powers of Arithmetic-Geometric Mean in the limit

Prove $\int_0^\infty \frac{dx}{\sqrt{(x^4+a^4)(x^4+b^4)}}=\frac{\pi}{2 \sqrt2 a b} ( \text{agm} (\frac{a+b}{2},\sqrt{\frac{a^2+b^2}{2}} ))^{-1}$

Calculate $\int_0^1\frac{\cos(\arctan x)}{\sqrt{x}}dx$

Principal value of complete elliptic integral of third kind

Principal value of Carlson elliptic integral $R_C$

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