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Fungrim entry: 0455b3

K(m)=0π/211msin2 ⁣(x)dxK(m) = \int_{0}^{\pi / 2} \frac{1}{\sqrt{1 - m \sin^{2}\!\left(x\right)}} \, dx
Assumptions:mC[1,)m \in \mathbb{C} \setminus \left[1, \infty\right)
TeX:
K(m) = \int_{0}^{\pi / 2} \frac{1}{\sqrt{1 - m \sin^{2}\!\left(x\right)}} \, dx

m \in \mathbb{C} \setminus \left[1, \infty\right)
Definitions:
Fungrim symbol Notation Short description
EllipticKK(m)K(m) Legendre complete elliptic integral of the first kind
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
Sqrtz\sqrt{z} Principal square root
Powab{a}^{b} Power
Sinsin(z)\sin(z) Sine
Piπ\pi The constant pi (3.14...)
CCC\mathbb{C} Complex numbers
ClosedOpenInterval[a,b)\left[a, b\right) Closed-open interval
Infinity\infty Positive infinity
Source code for this entry:
Entry(ID("0455b3"),
    Formula(Equal(EllipticK(m), Integral(Div(1, Sqrt(Sub(1, Mul(m, Pow(Sin(x), 2))))), For(x, 0, Div(Pi, 2))))),
    Variables(m),
    Assumptions(Element(m, SetMinus(CC, ClosedOpenInterval(1, Infinity)))))

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2021-03-15 19:12:00.328586 UTC