$K(m) = \int_{0}^{1} \frac{1}{\sqrt{1 - {x}^{2}} \sqrt{1 - m {x}^{2}}} \, dx$
Assumptions:$m \in \mathbb{C} \setminus \left[1, \infty\right)$
TeX:
K(m) = \int_{0}^{1} \frac{1}{\sqrt{1 - {x}^{2}} \sqrt{1 - m {x}^{2}}} \, dx

m \in \mathbb{C} \setminus \left[1, \infty\right)
Definitions:
Fungrim symbol Notation Short description
EllipticK$K(m)$ Legendre complete elliptic integral of the first kind
Integral$\int_{a}^{b} f(x) \, dx$ Integral
Sqrt$\sqrt{z}$ Principal square root
Pow${a}^{b}$ Power
CC$\mathbb{C}$ Complex numbers
ClosedOpenInterval$\left[a, b\right)$ Closed-open interval
Infinity$\infty$ Positive infinity
Source code for this entry:
Entry(ID("47dead"),
Formula(Equal(EllipticK(m), Integral(Div(1, Mul(Sqrt(Sub(1, Pow(x, 2))), Sqrt(Sub(1, Mul(m, Pow(x, 2)))))), For(x, 0, 1)))),
Variables(m),
Assumptions(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