# Fungrim entry: 2ff7e7

$E\!\left(\phi, m\right) = \int_{0}^{\phi} \sqrt{1 - m \sin^{2}\!\left(x\right)} \, dx$
Assumptions:$\phi \in \left[\frac{-\pi}{2}, \frac{\pi}{2}\right] \;\mathbin{\operatorname{and}}\; m \in \mathbb{C}$
TeX:
E\!\left(\phi, m\right) = \int_{0}^{\phi} \sqrt{1 - m \sin^{2}\!\left(x\right)} \, dx

\phi \in \left[\frac{-\pi}{2}, \frac{\pi}{2}\right] \;\mathbin{\operatorname{and}}\; m \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
IncompleteEllipticE$E\!\left(\phi, m\right)$ Legendre incomplete elliptic integral of the second kind
Integral$\int_{a}^{b} f(x) \, dx$ Integral
Sqrt$\sqrt{z}$ Principal square root
Pow${a}^{b}$ Power
Sin$\sin(z)$ Sine
ClosedInterval$\left[a, b\right]$ Closed interval
Pi$\pi$ The constant pi (3.14...)
CC$\mathbb{C}$ Complex numbers
Source code for this entry:
Entry(ID("2ff7e7"),
Formula(Equal(IncompleteEllipticE(phi, m), Integral(Sqrt(Sub(1, Mul(m, Pow(Sin(x), 2)))), For(x, 0, phi)))),
Variables(phi, m),
Assumptions(And(Element(phi, ClosedInterval(Div(Neg(Pi), 2), Div(Pi, 2))), Element(m, CC))))

## 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