How to derive elliptic integral of the first kind from $\int_{-\pi}^{\pi}\frac{1}{\sqrt{(\eta^{2}/2 - 3/2 -\cos 2\theta)^2 -4 \cos^2\theta}}\ d\theta$

score 275 · Answer 9 · 2023-03-24 11:30:08

275

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Evaluate $\int_{0}^{1} \frac{K(k)E(k)^2-\frac{\pi^3}{8} }{k} \text{d}k$ and $\int_{0}^{1} \frac{E(k)^3-\frac{\pi^3}{8} }{k} \text{d}k$

Published on 24 Mar 2023 - 11:30

score 50 · Answer 10 · 2023-03-24 19:25:04

50

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How to show that $\int_0^1 \frac{1}{\sqrt{x(1-x)(x+c) }}=\frac{2K(-\frac{1}{c})}{\sqrt{c}}$?

Published on 24 Mar 2023 - 19:25

score 56 · Answer 11 · 2023-03-28 14:12:56

56

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Expansion of elliptic integral close to a critical point

Published on 28 Mar 2023 - 14:12

score 155 · Answer 12 · 2026-03-25 17:37:33

155

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Integrals of Jacobi $\vartheta$ functions on the interval $[1,+\infty)$

Published on 25 Mar 2026 - 17:37

score 1k · Answer 13 · 2026-03-26 02:57:46

1k

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Prove $\int_{0}^{1} \frac{k^{\frac34}}{(1-k^2)^\frac38} K(k)\text{d}k=\frac{\pi^2}{12}\sqrt{5+\frac{1}{\sqrt{2} } }$

Published on 26 Mar 2026 - 2:57

score 376 · Answer 14 · 2023-04-30 06:20:23

376

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Evaluate $I(a,b)=\int_{0}^{1}k^a(1-k^2)^bK(k)\text{d}k$

Published on 30 Apr 2023 - 6:20

score 73 · Answer 15 · 2026-02-23 08:33:55

73

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Integrating $\int \frac{dx}{\sqrt{x^5+ax^3+bx^2+cx+d}}$.

Published on 23 Feb 2026 - 8:33

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How to derive elliptic integral of the first kind from $\int_{-\pi}^{\pi}\frac{1}{\sqrt{(\eta^{2}/2 - 3/2 -\cos 2\theta)^2 -4 \cos^2\theta}}\ d\theta$

Evaluate $\int_{0}^{1} \frac{K(k)E(k)^2-\frac{\pi^3}{8} }{k} \text{d}k$ and $\int_{0}^{1} \frac{E(k)^3-\frac{\pi^3}{8} }{k} \text{d}k$

How to show that $\int_0^1 \frac{1}{\sqrt{x(1-x)(x+c) }}=\frac{2K(-\frac{1}{c})}{\sqrt{c}}$?

Expansion of elliptic integral close to a critical point

Integrals of Jacobi $\vartheta$ functions on the interval $[1,+\infty)$

Prove $\int_{0}^{1} \frac{k^{\frac34}}{(1-k^2)^\frac38} K(k)\text{d}k=\frac{\pi^2}{12}\sqrt{5+\frac{1}{\sqrt{2} } }$

Evaluate $I(a,b)=\int_{0}^{1}k^a(1-k^2)^bK(k)\text{d}k$

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