# Questions tagged [elliptic-integrals]

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

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### Bauer's series for $\frac{1}{\pi}$

Recently, someone asked a question involving the expression $$\sum_{n=0}^{\infty} (-1)^n (4n+1) \left(\frac{(2n-1)!!}{(2n)!!}\right)^3$$ At first glance, I knew that the expression was the value of ...
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### Derive the use of the Jacobi Amplitude function with the nonlinear pendulum diffequation

I've been playing around with the pendulum differential equation ${\theta}''+\frac{g}{L}\sin{\theta}=0$, and have found many general solutions using the Jacobi Amplitude function $\text{am}(u,m)$ ...
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### On Ramanujan's fastest series.

Context With some effort we can show that Ramanujan's fastest series implies: \begin{align} \frac{8E(k_{58})K(k_{58})}{\pi^2}-\frac{aK^2(k_{58})}{\pi^2}=\frac{\sqrt{58}}{29\pi},\tag{1} \end{align} ...
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### Indefinite integral of elliptic integrals

Derivative of complete elliptic integrals, $E(k)$, $K(k)$, etc., are known. But, I don't know about their integrals. I tried to evaluate the indefinite integral $$\int_0^k K(k)dk\tag1$$ and ended up ...
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