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Legendre elliptic integrals

Table of contents: Definitions - Illustrations - Integral representations - Specific values - Functional equations - Representation by other functions - Representation of other functions

Definitions

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Symbol: EllipticK — K(m)K(m) — Legendre complete elliptic integral of the first kind
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Symbol: EllipticE — E(m)E(m) — Legendre complete elliptic integral of the second kind
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Symbol: EllipticPi — Π ⁣(n,m)\Pi\!\left(n, m\right) — Legendre complete elliptic integral of the third kind
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Symbol: IncompleteEllipticF — F ⁣(ϕ,m)F\!\left(\phi, m\right) — Legendre incomplete elliptic integral of the first kind
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Symbol: IncompleteEllipticE — E ⁣(ϕ,m)E\!\left(\phi, m\right) — Legendre incomplete elliptic integral of the second kind
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Symbol: IncompleteEllipticPi — Π ⁣(n,ϕ,m)\Pi\!\left(n, \phi, m\right) — Legendre incomplete elliptic integral of the third kind

Illustrations

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Image: Plot of K(m)K(m) on m∈[−2,2]m \in \left[-2, 2\right]
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Image: Plot of E(m)E(m) on m∈[−2,2]m \in \left[-2, 2\right]
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Image: Plot of F ⁣(ϕ,m)F\!\left(\phi, m\right) on ϕ∈[−2π,2π]\phi \in \left[-2 \pi, 2 \pi\right]
20d72c
Image: Plot of E ⁣(ϕ,m)E\!\left(\phi, m\right) on ϕ∈[−2π,2π]\phi \in \left[-2 \pi, 2 \pi\right]

Integral representations

Trigonometric forms of the complete integrals

0455b3
K(m)=∫0π/211−msin⁡2 ⁣(x) dxK(m) = \int_{0}^{\pi / 2} \frac{1}{\sqrt{1 - m \sin^{2}\!\left(x\right)}} \, dx
190843
E(m)=∫0π/21−msin⁡2 ⁣(x) dxE(m) = \int_{0}^{\pi / 2} \sqrt{1 - m \sin^{2}\!\left(x\right)} \, dx
83a535
Π ⁣(n,m)=∫0π/21(1−nsin⁡2 ⁣(x))1−msin⁡2 ⁣(x) dx\Pi\!\left(n, m\right) = \int_{0}^{\pi / 2} \frac{1}{\left(1 - n \sin^{2}\!\left(x\right)\right) \sqrt{1 - m \sin^{2}\!\left(x\right)}} \, dx

Algebraic forms of the complete integrals

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K(m)=∫0111−x21−mx2 dxK(m) = \int_{0}^{1} \frac{1}{\sqrt{1 - {x}^{2}} \sqrt{1 - m {x}^{2}}} \, dx
fa8666
E(m)=∫011−mx21−x2 dxE(m) = \int_{0}^{1} \frac{\sqrt{1 - m {x}^{2}}}{\sqrt{1 - {x}^{2}}} \, dx
c10014
Π ⁣(n,m)=∫011(1−nx2)1−x21−mx2 dx\Pi\!\left(n, m\right) = \int_{0}^{1} \frac{1}{\left(1 - n {x}^{2}\right) \sqrt{1 - {x}^{2}} \sqrt{1 - m {x}^{2}}} \, dx
7cd257
K(m)=∫1∞1x2−1x2−m dxK(m) = \int_{1}^{\infty} \frac{1}{\sqrt{{x}^{2} - 1} \sqrt{{x}^{2} - m}} \, dx

Trigonometric forms of the incomplete integrals

81fb10
F ⁣(ϕ,m)=∫0ϕ11−msin⁡2 ⁣(x) dxF\!\left(\phi, m\right) = \int_{0}^{\phi} \frac{1}{\sqrt{1 - m \sin^{2}\!\left(x\right)}} \, dx
2ff7e7
E ⁣(ϕ,m)=∫0ϕ1−msin⁡2 ⁣(x) dxE\!\left(\phi, m\right) = \int_{0}^{\phi} \sqrt{1 - m \sin^{2}\!\left(x\right)} \, dx
60f858
Π ⁣(n,ϕ,m)=∫0ϕ1(1−nsin⁡2 ⁣(x))1−msin⁡2 ⁣(x) dx\Pi\!\left(n, \phi, m\right) = \int_{0}^{\phi} \frac{1}{\left(1 - n \sin^{2}\!\left(x\right)\right) \sqrt{1 - m \sin^{2}\!\left(x\right)}} \, dx

Algebraic forms of the incomplete integrals

33ee4a
F ⁣(ϕ,m)=∫0sin⁡(ϕ)11−x21−mx2 dxF\!\left(\phi, m\right) = \int_{0}^{\sin(\phi)} \frac{1}{\sqrt{1 - {x}^{2}} \sqrt{1 - m {x}^{2}}} \, dx
5e869b
E ⁣(ϕ,m)=∫0sin⁡(ϕ)1−mx21−x2 dxE\!\left(\phi, m\right) = \int_{0}^{\sin(\phi)} \frac{\sqrt{1 - m {x}^{2}}}{\sqrt{1 - {x}^{2}}} \, dx
06223c
Π ⁣(n,ϕ,m)=∫0sin⁡(ϕ)1(1−nx2)1−x21−mx2 dx\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

Specific values

Complete elliptic integrals

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K(0)=π2K(0) = \frac{\pi}{2}
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E(0)=π2E(0) = \frac{\pi}{2}
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K(1)=∞K(1) = \infty
958a3f
E(1)=1E(1) = 1
afb22a
K(−1)=(Γ ⁣(14))242πK(-1) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}}
cc22bf
K ⁣(12)=(Γ ⁣(14))24πK\!\left(\frac{1}{2}\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{\pi}}
630eca
K(2)=(Γ ⁣(14))242π(1−i)K(2) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}} \left(1 - i\right)
9f3474
E(−1)=2((Γ ⁣(14))28π+π3/2(Γ ⁣(14))2)E(-1) = \sqrt{2} \left(\frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{8 \sqrt{\pi}} + \frac{{\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}\right)
3b272e
E ⁣(12)=(Γ ⁣(14))28π+π3/2(Γ ⁣(14))2E\!\left(\frac{1}{2}\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{8 \sqrt{\pi}} + \frac{{\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}
5d2c01
E(2)=2π3/2(Γ ⁣(14))2(1+i)E(2) = \frac{\sqrt{2} {\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}} \left(1 + i\right)
2991b5
K ⁣((3−22)2)=(2+2)(Γ ⁣(14))216πK\!\left({\left(3 - 2 \sqrt{2}\right)}^{2}\right) = \frac{\left(2 + \sqrt{2}\right) {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{16 \sqrt{\pi}}
4b040d
K ⁣(4−328)=(Γ ⁣(14))24⋅21/4πK\!\left(\frac{4 - 3 \sqrt{2}}{8}\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \cdot {2}^{1 / 4} \sqrt{\pi}}
0abbe1
K ⁣(1+3i2)=eiπ/12⋅31/4(Γ ⁣(13))327/3πK\!\left(\frac{1 + \sqrt{3} i}{2}\right) = \frac{{e}^{i \pi / 12} \cdot {3}^{1 / 4} {\left(\Gamma\!\left(\frac{1}{3}\right)\right)}^{3}}{{2}^{7 / 3} \pi}
175b7a
K ⁣(1−3i2)=e−iπ/12⋅31/4(Γ ⁣(13))327/3πK\!\left(\frac{1 - \sqrt{3} i}{2}\right) = \frac{{e}^{-i \pi / 12} \cdot {3}^{1 / 4} {\left(\Gamma\!\left(\frac{1}{3}\right)\right)}^{3}}{{2}^{7 / 3} \pi}
b95ffa
K ⁣(43−7)=3+23(Γ ⁣(13))3210/3πK\!\left(4 \sqrt{3} - 7\right) = \frac{\sqrt{3 + 2 \sqrt{3}} {\left(\Gamma\!\left(\frac{1}{3}\right)\right)}^{3}}{{2}^{10 / 3} \pi}
40a376
K ⁣(12−34)=31/4(Γ ⁣(13))34⋅21/3πK\!\left(\frac{1}{2} - \frac{\sqrt{3}}{4}\right) = \frac{{3}^{1 / 4} {\left(\Gamma\!\left(\frac{1}{3}\right)\right)}^{3}}{4 \cdot {2}^{1 / 3} \pi}
618a54
Π ⁣(0,0)=π2\Pi\!\left(0, 0\right) = \frac{\pi}{2}
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Π ⁣(0,1)=∞\Pi\!\left(0, 1\right) = \infty
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Π ⁣(1,0)=∞~\Pi\!\left(1, 0\right) = {\tilde \infty}
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Π ⁣(0,12)=(Γ ⁣(14))24π\Pi\!\left(0, \frac{1}{2}\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{\pi}}
124d02
Π ⁣(12,0)=π22\Pi\!\left(\frac{1}{2}, 0\right) = \frac{\pi \sqrt{2}}{2}
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Π ⁣(12,12)=(Γ ⁣(14))24π+2π3/2(Γ ⁣(14))2\Pi\!\left(\frac{1}{2}, \frac{1}{2}\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{\pi}} + \frac{2 {\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}
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Π ⁣(1,m)=∞~\Pi\!\left(1, m\right) = {\tilde \infty}
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Π ⁣(n,1)={(1−n)−1∞,n≠1∞~,n=1\Pi\!\left(n, 1\right) = \begin{cases} {\left(1 - n\right)}^{-1} \infty, & n \ne 1\\{\tilde \infty}, & n = 1\\ \end{cases}
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Π ⁣(n,0)=π21−n\Pi\!\left(n, 0\right) = \frac{\pi}{2 \sqrt{1 - n}}
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Π ⁣(0,m)=K(m)\Pi\!\left(0, m\right) = K(m)
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Π ⁣(m,m)=E(m)1−m\Pi\!\left(m, m\right) = \frac{E(m)}{1 - m}

Incomplete integral of the first kind

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F ⁣(0,0)=0F\!\left(0, 0\right) = 0
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F ⁣(0,m)=0F\!\left(0, m\right) = 0
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F ⁣(ϕ,0)=ϕF\!\left(\phi, 0\right) = \phi
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F ⁣(π2,m)=K(m)F\!\left(\frac{\pi}{2}, m\right) = K(m)
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F ⁣(−π2,m)=−K(m)F\!\left(\frac{-\pi}{2}, m\right) = -K(m)
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F ⁣(πk2,m)=kK(m)F\!\left(\frac{\pi k}{2}, m\right) = k K(m)
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F ⁣(π2,0)=π2F\!\left(\frac{\pi}{2}, 0\right) = \frac{\pi}{2}
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F ⁣(π2,−1)=(Γ ⁣(14))242πF\!\left(\frac{\pi}{2}, -1\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}}
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F ⁣(π2,1)=∞F\!\left(\frac{\pi}{2}, 1\right) = \infty
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F ⁣(−π2,1)=−∞F\!\left(\frac{-\pi}{2}, 1\right) = -\infty
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F ⁣(π3,1)=log⁡ ⁣(2+3)F\!\left(\frac{\pi}{3}, 1\right) = \log\!\left(2 + \sqrt{3}\right)
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F ⁣(π4,1)=log⁡ ⁣(1+2)F\!\left(\frac{\pi}{4}, 1\right) = \log\!\left(1 + \sqrt{2}\right)
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F ⁣(π6,1)=log⁡(3)2F\!\left(\frac{\pi}{6}, 1\right) = \frac{\log(3)}{2}
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F ⁣(ϕ,1)={log⁡ ⁣(1+sin⁡(ϕ)cos⁡(ϕ)),−π2≤Re⁡(ϕ)≤π2  and⁡  ϕ∉{−π2,π2}sgn⁡(ϕ)∞,ϕ∈{−π2,π2}∞~,otherwiseF\!\left(\phi, 1\right) = \begin{cases} \log\!\left(\frac{1 + \sin(\phi)}{\cos(\phi)}\right), & \frac{-\pi}{2} \le \operatorname{Re}(\phi) \le \frac{\pi}{2} \;\mathbin{\operatorname{and}}\; \phi \notin \left\{\frac{-\pi}{2}, \frac{\pi}{2}\right\}\\\operatorname{sgn}(\phi) \infty, & \phi \in \left\{\frac{-\pi}{2}, \frac{\pi}{2}\right\}\\{\tilde \infty}, & \text{otherwise}\\ \end{cases}
8b4be6
F ⁣(π4,2)=2(Γ ⁣(14))28πF\!\left(\frac{\pi}{4}, 2\right) = \frac{\sqrt{2} {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{8 \sqrt{\pi}}
aac129
F ⁣(π6,4)=K ⁣(14)2F\!\left(\frac{\pi}{6}, 4\right) = \frac{K\!\left(\frac{1}{4}\right)}{2}
087a7c
F ⁣(asin⁡ ⁣(1m),m)=K ⁣(1m)mF\!\left(\operatorname{asin}\!\left(\frac{1}{\sqrt{m}}\right), m\right) = \frac{K\!\left(\frac{1}{m}\right)}{\sqrt{m}}

Incomplete integral of the second kind

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E ⁣(0,0)=0E\!\left(0, 0\right) = 0
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E ⁣(0,m)=0E\!\left(0, m\right) = 0
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E ⁣(ϕ,0)=ϕE\!\left(\phi, 0\right) = \phi
1b881e
E ⁣(π2,m)=E(m)E\!\left(\frac{\pi}{2}, m\right) = E(m)
2ef763
E ⁣(−π2,m)=−E(m)E\!\left(\frac{-\pi}{2}, m\right) = -E(m)
a14442
E ⁣(πk2,m)=kE(m)E\!\left(\frac{\pi k}{2}, m\right) = k E(m)
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E ⁣(ϕ,1)=sin⁡(ϕ)E\!\left(\phi, 1\right) = \sin(\phi)
f35a37
E ⁣(ϕ,1)=(−1)⌊Re⁡(ϕ)/π+1/2⌋sin⁡(ϕ)+2⌊Re⁡(ϕ)π+12⌋E\!\left(\phi, 1\right) = {\left(-1\right)}^{\left\lfloor \operatorname{Re}(\phi) / \pi + 1 / 2 \right\rfloor} \sin(\phi) + 2 \left\lfloor \frac{\operatorname{Re}(\phi)}{\pi} + \frac{1}{2} \right\rfloor
51a946
E ⁣(π2,0)=π2E\!\left(\frac{\pi}{2}, 0\right) = \frac{\pi}{2}
2573ba
E ⁣(π2,−1)=2((Γ ⁣(14))28π+π3/2(Γ ⁣(14))2)E\!\left(\frac{\pi}{2}, -1\right) = \sqrt{2} \left(\frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{8 \sqrt{\pi}} + \frac{{\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}\right)
b62aae
E ⁣(π2,1)=1E\!\left(\frac{\pi}{2}, 1\right) = 1
dec0d2
E ⁣(−π2,1)=−1E\!\left(\frac{-\pi}{2}, 1\right) = -1
2245df
E ⁣(πk2,1)=kE\!\left(\frac{\pi k}{2}, 1\right) = k
3aed02
E ⁣(π3,1)=32E\!\left(\frac{\pi}{3}, 1\right) = \frac{\sqrt{3}}{2}
d88dd1
E ⁣(π6,1)=12E\!\left(\frac{\pi}{6}, 1\right) = \frac{1}{2}
4dabda
E ⁣(π4,2)=2π3/2(Γ ⁣(14))2E\!\left(\frac{\pi}{4}, 2\right) = \frac{\sqrt{2} {\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}
eba27c
E ⁣(π6,4)=2E ⁣(14)−32K ⁣(14)E\!\left(\frac{\pi}{6}, 4\right) = 2 E\!\left(\frac{1}{4}\right) - \frac{3}{2} K\!\left(\frac{1}{4}\right)
f0bcb5
E ⁣(asin⁡ ⁣(1m),m)=m(E ⁣(1m)−(1−1m)K ⁣(1m))E\!\left(\operatorname{asin}\!\left(\frac{1}{\sqrt{m}}\right), m\right) = \sqrt{m} \left(E\!\left(\frac{1}{m}\right) - \left(1 - \frac{1}{m}\right) K\!\left(\frac{1}{m}\right)\right)

Functional equations

Conjugate symmetry

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K ⁣(m‾)=K(m)‾K\!\left(\overline{m}\right) = \overline{K(m)}
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E ⁣(m‾)=E(m)‾E\!\left(\overline{m}\right) = \overline{E(m)}

Odd symmetry

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F ⁣(−ϕ,m)=−F ⁣(ϕ,m)F\!\left(-\phi, m\right) = -F\!\left(\phi, m\right)
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E ⁣(−ϕ,m)=−E ⁣(ϕ,m)E\!\left(-\phi, m\right) = -E\!\left(\phi, m\right)
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Π ⁣(n,−ϕ,m)=−Π ⁣(n,ϕ,m)\Pi\!\left(n, -\phi, m\right) = -\Pi\!\left(n, \phi, m\right)

Quasi-periodicity

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F ⁣(ϕ+kπ,m)=F ⁣(ϕ,m)+2kK(m)F\!\left(\phi + k \pi, m\right) = F\!\left(\phi, m\right) + 2 k K(m)
c28288
E ⁣(ϕ+kπ,m)=E ⁣(ϕ,m)+2kE(m)E\!\left(\phi + k \pi, m\right) = E\!\left(\phi, m\right) + 2 k E(m)
5f84d9
Π ⁣(n,ϕ+kπ,m)=Π ⁣(n,ϕ,m)+2kΠ ⁣(n,m)\Pi\!\left(n, \phi + k \pi, m\right) = \Pi\!\left(n, \phi, m\right) + 2 k \Pi\!\left(n, m\right)

Representation by other functions

Hypergeometric functions

b760d1
K(m)=π2 2F1 ⁣(12,12,1,m)K(m) = \frac{\pi}{2} \,{}_2F_1\!\left(\frac{1}{2}, \frac{1}{2}, 1, m\right)
16d2e1
E(m)=π2 2F1 ⁣(−12,12,1,m)E(m) = \frac{\pi}{2} \,{}_2F_1\!\left(-\frac{1}{2}, \frac{1}{2}, 1, m\right)
752619
2E(m)−K(m)=π2 2F1 ⁣(−12,32,1,m)2 E(m) - K(m) = \frac{\pi}{2} \,{}_2F_1\!\left(-\frac{1}{2}, \frac{3}{2}, 1, m\right)

Arithmetic-geometric mean

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K(m)=π2agm⁡ ⁣(1,1−m)K(m) = \frac{\pi}{2 \operatorname{agm}\!\left(1, \sqrt{1 - m}\right)}

Carlson symmetric elliptic integrals

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K(m)=RF ⁣(0,1−m,1)K(m) = R_F\!\left(0, 1 - m, 1\right)
6520e7
E(m)=2RG ⁣(0,1−m,1)E(m) = 2 R_G\!\left(0, 1 - m, 1\right)
9ccaef
Π ⁣(n,m)=RF ⁣(0,1−m,1)+n3RJ ⁣(0,1−m,1,1−n)\Pi\!\left(n, m\right) = R_F\!\left(0, 1 - m, 1\right) + \frac{n}{3} R_J\!\left(0, 1 - m, 1, 1 - n\right)
41cf8e