# Questions tagged [elliptic-integrals]

Questions on elliptic integrals, integrals that involve the square root of a cubic or quartic polynomial.

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### Given algebraic $a$, find the closed form of $\int_0^a \dfrac{dx}{\sqrt{1-x^4}}$

Let $$A=\int_0^1 \dfrac{dx}{\sqrt{1-x^4}}.$$ Given an algebraic number $0\le a\le 1$, can we determine if there exists a rational number $b$ such that $$\int_0^a \dfrac{dx}{\sqrt{1-x^4}}=Ab?$$ If so, ...
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### Prove $\int_{0}^{1}\frac1k K(k)\ln\left[\frac{\left(1+k \right)^3}{1-k} \right]\text{d}k=\frac{\pi^3}{4}$

Is it possible to show $$\int_{0}^{1}\frac{K(k)\ln\left[\tfrac{\left ( 1+k \right)^3}{1-k} \right] }{k} \text{d}k=\frac{\pi^3}{4}\;\;?$$ where $K(k)$ is the complete elliptic integral of the first ...
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### Solving the Elliptic Integral of the First Kind with Imaginary Moduli

The Complete Elliptic Integral of the First Kind is defined as the following: $$K(m)=\int_0^{\frac{\pi}{2}}\frac{dx}{\sqrt{1-m\sin^2(x)}}$$ Given this definition, I would like to solve the following ...
1 vote
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### Requesting clarification of Galois' last letter

I am reading Galois' last letter to Auguste Chevalier before the infamous duel. https://www.ias.ac.in/article/fulltext/reso/004/10/0093-0100 What is the notation H' for a group H supposed to represent ...
1 vote
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### Aircraft Wing Analysis - Elliptical Pressure Distribution

I'm having great difficulty with generating the shear and bending moment diagrams for a wingspan. I am using the elliptical pressure distribution equation; however, this problem doesn't consider a ...
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