# Fungrim entry: 33ee4a

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

\phi \in \left[\frac{-\pi}{2}, \frac{\pi}{2}\right] \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} \setminus \left[1, \infty\right)
Definitions:
Fungrim symbol Notation Short description
IncompleteEllipticF$F\!\left(\phi, m\right)$ Legendre incomplete elliptic integral of the first 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
ClosedOpenInterval$\left[a, b\right)$ Closed-open interval
Infinity$\infty$ Positive infinity
Source code for this entry:
Entry(ID("33ee4a"),
Formula(Equal(IncompleteEllipticF(phi, m), Integral(Div(1, Mul(Sqrt(Sub(1, Pow(x, 2))), Sqrt(Sub(1, Mul(m, Pow(x, 2)))))), For(x, 0, Sin(phi))))),
Variables(phi, m),
Assumptions(And(Element(phi, ClosedInterval(Div(Neg(Pi), 2), Div(Pi, 2))), Element(m, SetMinus(CC, ClosedOpenInterval(1, Infinity))))))

## 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