New answers tagged elliptic-integrals
1
vote
$\int_0^1 \sqrt{\frac{2-x^4}{1-x^4}}\, dx$ in terms of the gamma function or elliptic integrals
Too long for comment and not really an answer to the central question, but here is how we can recover the hypergeometric result:
$$\begin{align*}
I &= \int_0^1 \sqrt{\frac{2-x^4}{1-x^4}} \, dx \\
&...
- 21.2k
1
vote
Exact solution to integral (Keplerian free fall time)
We have the ODE with $k=G M$
$$r''(t)=-\frac{k}{r(t)^2}$$
We multiply the factor $2r'(t)$ on both sides.
$$2r'(t)r''(t)=-\frac{2k}{r(t)^2}r'(t)$$
No we integrate the equation over time
$$r'(t)^2=\int -...
- 1,184
6
votes
Accepted
Is this a 'standard' elliptic integral?
The first integral is a third type elliptic one, of the form
$$\frac1{(1+\mu)^{3/2}}\int\dfrac{d\alpha}{\left(1-\frac{2\mu}{1+\mu}\cos^2\frac\alpha2\right)\sqrt{(1-\frac{2\mu}{1+\mu}\cos^2\frac\alpha2)...
- 3,510
10
votes
Is this a 'standard' elliptic integral?
Maple says
$$
\int_{0}^{\pi}\! \left( 1-\mu\,\cos \left( \alpha \right) \right) ^{-
{\frac{3}{2}}}\,{\rm d}\alpha=-2\,{\frac {1}{\sqrt {\mu+1} \left( -1+
\mu \right) }{E} \left( {\frac {\sqrt {2}\...
- 114k
0
votes
An Elliptic Curve has 3 real roots if and only if $\frac{\omega_2}{\omega_1}$ is purely imaginary
I think that for the other directon OP's approach is good.
For the other direction, WLOG, we may assume that $e_3<e_2<e_1$. Then, by direct calculation, we can find that
$$\omega_1 = \int_{\...
- 12.5k
Top 50 recent answers are included