Fungrim entry: 06223c

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

\phi \in \left[\frac{-\pi}{2}, \frac{\pi}{2}\right] \;\mathbin{\operatorname{and}}\; n \in \left(-\infty, 1\right) \;\mathbin{\operatorname{and}}\; m \in \left(-\infty, 1\right)
Definitions:
Fungrim symbol Notation Short description
IncompleteEllipticPi$\Pi\!\left(n, \phi, m\right)$ Legendre incomplete elliptic integral of the third kind
Integral$\int_{a}^{b} f(x) \, dx$ Integral
Pow${a}^{b}$ Power
Sqrt$\sqrt{z}$ Principal square root
Sin$\sin(z)$ Sine
ClosedInterval$\left[a, b\right]$ Closed interval
Pi$\pi$ The constant pi (3.14...)
OpenInterval$\left(a, b\right)$ Open interval
Infinity$\infty$ Positive infinity
Source code for this entry:
Entry(ID("06223c"),
Formula(Equal(IncompleteEllipticPi(n, phi, m), Integral(Div(1, Mul(Mul(Sub(1, Mul(n, Pow(x, 2))), Sqrt(Sub(1, Pow(x, 2)))), Sqrt(Sub(1, Mul(m, Pow(x, 2)))))), For(x, 0, Sin(phi))))),
Variables(n, phi, m),
Assumptions(And(Element(phi, ClosedInterval(Div(Neg(Pi), 2), Div(Pi, 2))), Element(n, OpenInterval(Neg(Infinity), 1)), Element(m, OpenInterval(Neg(Infinity), 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