# Fungrim entry: f0bcb5

$E\!\left(\operatorname{asin}\!\left(\frac{1}{\sqrt{m}}\right), m\right) = \sqrt{m} \left(E\!\left(\frac{1}{m}\right) - \left(1 - \frac{1}{m}\right) K\!\left(\frac{1}{m}\right)\right)$
Assumptions:$m \in \mathbb{C} \setminus \left\{0, 1\right\}$
TeX:
E\!\left(\operatorname{asin}\!\left(\frac{1}{\sqrt{m}}\right), m\right) = \sqrt{m} \left(E\!\left(\frac{1}{m}\right) - \left(1 - \frac{1}{m}\right) K\!\left(\frac{1}{m}\right)\right)

m \in \mathbb{C} \setminus \left\{0, 1\right\}
Definitions:
Fungrim symbol Notation Short description
IncompleteEllipticE$E\!\left(\phi, m\right)$ Legendre incomplete elliptic integral of the second kind
Sqrt$\sqrt{z}$ Principal square root
EllipticE$E(m)$ Legendre complete elliptic integral of the second kind
EllipticK$K(m)$ Legendre complete elliptic integral of the first kind
CC$\mathbb{C}$ Complex numbers
Source code for this entry:
Entry(ID("f0bcb5"),
Formula(Equal(IncompleteEllipticE(Asin(Div(1, Sqrt(m))), m), Mul(Sqrt(m), Sub(EllipticE(Div(1, m)), Mul(Sub(1, Div(1, m)), EllipticK(Div(1, m))))))),
Variables(m),
Assumptions(Element(m, SetMinus(CC, Set(0, 1)))))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC