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Carlson symmetric elliptic integrals

Table of contents: Definitions - Illustrations - Integral representations - Connection formulas - Symmetry - Scale invariance - Domain - Representation of other functions - Specific values - Functional equations - Derivatives and differential equations - Series representations - Bounds and inequalities

Definitions

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Symbol: CarlsonRF RF ⁣(x,y,z)R_F\!\left(x, y, z\right) Carlson symmetric elliptic integral of the first kind
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Symbol: CarlsonRG RG ⁣(x,y,z)R_G\!\left(x, y, z\right) Carlson symmetric elliptic integral of the second kind
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Symbol: CarlsonRJ RJ ⁣(x,y,z,w)R_J\!\left(x, y, z, w\right) Carlson symmetric elliptic integral of the third kind
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Symbol: CarlsonRD RD ⁣(x,y,z)R_D\!\left(x, y, z\right) Degenerate Carlson symmetric elliptic integral of the third kind
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Symbol: CarlsonRC RC ⁣(x,y)R_C\!\left(x, y\right) Degenerate Carlson symmetric elliptic integral of the first kind

Illustrations

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Image: Plot of RF ⁣(x,y,1)R_F\!\left(x, y, 1\right) on x[0,4]x \in \left[0, 4\right] for y{0.1,1,10}y \in \left\{0.1, 1, 10\right\}
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Image: Plot of RG ⁣(x,y,1)R_G\!\left(x, y, 1\right) on x[0,4]x \in \left[0, 4\right] for y{0.1,1,10}y \in \left\{0.1, 1, 10\right\}
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Image: X-ray of RF ⁣(z,1,2)R_F\!\left(z, 1, 2\right) on z[4,4]+[4,4]iz \in \left[-4, 4\right] + \left[-4, 4\right] i

Integral representations

Defining algebraic integrals

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RF ⁣(x,y,z)=1201t+xt+yt+zdtR_F\!\left(x, y, z\right) = \frac{1}{2} \int_{0}^{\infty} \frac{1}{\sqrt{t + x} \sqrt{t + y} \sqrt{t + z}} \, dt
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RG ⁣(x,y,z)=140tt+xt+yt+z(xt+x+yt+y+zt+z)dtR_G\!\left(x, y, z\right) = \frac{1}{4} \int_{0}^{\infty} \frac{t}{\sqrt{t + x} \sqrt{t + y} \sqrt{t + z}} \left(\frac{x}{t + x} + \frac{y}{t + y} + \frac{z}{t + z}\right) \, dt
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RJ ⁣(x,y,z,w)=3201(t+w)t+xt+yt+zdtR_J\!\left(x, y, z, w\right) = \frac{3}{2} \int_{0}^{\infty} \frac{1}{\left(t + w\right) \sqrt{t + x} \sqrt{t + y} \sqrt{t + z}} \, dt
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RD ⁣(x,y,z)=3201(t+x)(t+y)(t+z)3/2dtR_D\!\left(x, y, z\right) = \frac{3}{2} \int_{0}^{\infty} \frac{1}{\left(t + x\right) \left(t + y\right) {\left(t + z\right)}^{3 / 2}} \, dt
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RC ⁣(x,y)=1201(t+y)t+xdtR_C\!\left(x, y\right) = \frac{1}{2} \int_{0}^{\infty} \frac{1}{\left(t + y\right) \sqrt{t + x}} \, dt

Trigonometric integrals

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RF ⁣(0,y,z)=0π/21ycos2 ⁣(θ)+zsin2 ⁣(θ)dθR_F\!\left(0, y, z\right) = \int_{0}^{\pi / 2} \frac{1}{\sqrt{y \cos^{2}\!\left(\theta\right) + z \sin^{2}\!\left(\theta\right)}} \, d\theta
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RG ⁣(0,y,z)=120π/2ycos2 ⁣(θ)+zsin2 ⁣(θ)dθR_G\!\left(0, y, z\right) = \frac{1}{2} \int_{0}^{\pi / 2} \sqrt{y \cos^{2}\!\left(\theta\right) + z \sin^{2}\!\left(\theta\right)} \, d\theta
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RD ⁣(0,y,z)=30π/2sin2 ⁣(θ)(ycos2 ⁣(θ)+zsin2 ⁣(θ))3/2dθR_D\!\left(0, y, z\right) = 3 \int_{0}^{\pi / 2} \frac{\sin^{2}\!\left(\theta\right)}{{\left(y \cos^{2}\!\left(\theta\right) + z \sin^{2}\!\left(\theta\right)\right)}^{3 / 2}} \, d\theta
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RF ⁣(x,y,z)=14π02π0πsin(θ)xsin2 ⁣(θ)cos2 ⁣(ϕ)+ysin2 ⁣(θ)sin2 ⁣(ϕ)+zcos2 ⁣(θ)dθdϕR_F\!\left(x, y, z\right) = \frac{1}{4 \pi} \int_{0}^{2 \pi} \int_{0}^{\pi} \frac{\sin(\theta)}{\sqrt{x \sin^{2}\!\left(\theta\right) \cos^{2}\!\left(\phi\right) + y \sin^{2}\!\left(\theta\right) \sin^{2}\!\left(\phi\right) + z \cos^{2}\!\left(\theta\right)}} \, d\theta \, d\phi
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RG ⁣(x,y,z)=14π02π0πxsin2 ⁣(θ)cos2 ⁣(ϕ)+ysin2 ⁣(θ)sin2 ⁣(ϕ)+zcos2 ⁣(θ)sin(θ)dθdϕR_G\!\left(x, y, z\right) = \frac{1}{4 \pi} \int_{0}^{2 \pi} \int_{0}^{\pi} \sqrt{x \sin^{2}\!\left(\theta\right) \cos^{2}\!\left(\phi\right) + y \sin^{2}\!\left(\theta\right) \sin^{2}\!\left(\phi\right) + z \cos^{2}\!\left(\theta\right)} \sin(\theta) \, d\theta \, d\phi

Connection formulas

Degenerate cases

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RC ⁣(x,y)=RF ⁣(x,y,y)R_C\!\left(x, y\right) = R_F\!\left(x, y, y\right)
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RD ⁣(x,y,z)=RJ ⁣(x,y,z,z)R_D\!\left(x, y, z\right) = R_J\!\left(x, y, z, z\right)

Connection formula for RF, RG and RD

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2RG ⁣(x,y,z)=zRF ⁣(x,y,z)(xz)(yz)3RD ⁣(x,y,z)+xyz2 R_G\!\left(x, y, z\right) = z R_F\!\left(x, y, z\right) - \frac{\left(x - z\right) \left(y - z\right)}{3} R_D\!\left(x, y, z\right) + \frac{\sqrt{x} \sqrt{y}}{\sqrt{z}}

Symmetry

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RF ⁣(x,y,z)=RF ⁣(x,z,y)=RF ⁣(y,x,z)=RF ⁣(y,z,x)=RF ⁣(z,x,y)=RF ⁣(z,y,x)R_F\!\left(x, y, z\right) = R_F\!\left(x, z, y\right) = R_F\!\left(y, x, z\right) = R_F\!\left(y, z, x\right) = R_F\!\left(z, x, y\right) = R_F\!\left(z, y, x\right)
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RG ⁣(x,y,z)=RG ⁣(x,z,y)=RG ⁣(y,x,z)=RG ⁣(y,z,x)=RG ⁣(z,x,y)=RG ⁣(z,y,x)R_G\!\left(x, y, z\right) = R_G\!\left(x, z, y\right) = R_G\!\left(y, x, z\right) = R_G\!\left(y, z, x\right) = R_G\!\left(z, x, y\right) = R_G\!\left(z, y, x\right)
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RJ ⁣(x,y,z,w)=RJ ⁣(x,z,y,w)=RJ ⁣(y,x,z,w)=RJ ⁣(y,z,x,w)=RJ ⁣(z,x,y,w)=RJ ⁣(z,y,x,w)R_J\!\left(x, y, z, w\right) = R_J\!\left(x, z, y, w\right) = R_J\!\left(y, x, z, w\right) = R_J\!\left(y, z, x, w\right) = R_J\!\left(z, x, y, w\right) = R_J\!\left(z, y, x, w\right)
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RD ⁣(x,y,z)=RD ⁣(y,x,z)R_D\!\left(x, y, z\right) = R_D\!\left(y, x, z\right)

Scale invariance

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RF ⁣(λx,λy,λz)=λ1/2RF ⁣(x,y,z)R_F\!\left(\lambda x, \lambda y, \lambda z\right) = {\lambda}^{-1 / 2} R_F\!\left(x, y, z\right)
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RG ⁣(λx,λy,λz)=λ1/2RG ⁣(x,y,z)R_G\!\left(\lambda x, \lambda y, \lambda z\right) = {\lambda}^{1 / 2} R_G\!\left(x, y, z\right)
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RJ ⁣(λx,λy,λz,λw)=λ3/2RJ ⁣(x,y,z,w)R_J\!\left(\lambda x, \lambda y, \lambda z, \lambda w\right) = {\lambda}^{-3 / 2} R_J\!\left(x, y, z, w\right)
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RD ⁣(λx,λy,λz)=λ3/2RD ⁣(x,y,z)R_D\!\left(\lambda x, \lambda y, \lambda z\right) = {\lambda}^{-3 / 2} R_D\!\left(x, y, z\right)
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RC ⁣(λx,λy)=λ1/2RC ⁣(x,y)R_C\!\left(\lambda x, \lambda y\right) = {\lambda}^{-1 / 2} R_C\!\left(x, y\right)

Domain

Complex variables

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(xC  and  yC  and  zC  and  ((x0  and  y0)  or  (x0  and  z0)  or  (y0  and  z0)))    (RF ⁣(x,y,z)C)\left(x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(\left(x \ne 0 \;\mathbin{\operatorname{and}}\; y \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(x \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(y \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right)\right)\right) \implies \left(R_F\!\left(x, y, z\right) \in \mathbb{C}\right)
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(xC  and  yC  and  zC)    (RG ⁣(x,y,z)C)\left(x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}\right) \implies \left(R_G\!\left(x, y, z\right) \in \mathbb{C}\right)
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(xC  and  yC  and  zC  and  wC{0}  and  ((x0  and  y0)  or  (x0  and  z0)  or  (y0  and  z0)))    (RJ ⁣(x,y,z,w)C)\left(x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; w \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; \left(\left(x \ne 0 \;\mathbin{\operatorname{and}}\; y \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(x \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(y \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right)\right)\right) \implies \left(R_J\!\left(x, y, z, w\right) \in \mathbb{C}\right)
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(xC  and  yC  and  zC{0}  and  (x0  or  y0))    (RD ⁣(x,y,z)C)\left(x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; \left(x \ne 0 \;\mathbin{\operatorname{or}}\; y \ne 0\right)\right) \implies \left(R_D\!\left(x, y, z\right) \in \mathbb{C}\right)
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(xC  and  yC{0})    (RC ⁣(x,y)C)\left(x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \setminus \left\{0\right\}\right) \implies \left(R_C\!\left(x, y\right) \in \mathbb{C}\right)

Real variables

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(x[0,)  and  y[0,)  and  z[0,)  and  ((x0  and  y0)  or  (x0  and  z0)  or  (y0  and  z0)))    (RF ⁣(x,y,z)(0,))\left(x \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; z \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; \left(\left(x \ne 0 \;\mathbin{\operatorname{and}}\; y \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(x \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(y \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right)\right)\right) \implies \left(R_F\!\left(x, y, z\right) \in \left(0, \infty\right)\right)
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(x[0,)  and  y[0,)  and  z[0,))    (RG ⁣(x,y,z)[0,))\left(x \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; z \in \left[0, \infty\right)\right) \implies \left(R_G\!\left(x, y, z\right) \in \left[0, \infty\right)\right)
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(x[0,)  and  y[0,)  and  z[0,)  and  w(0,)  and  ((x0  and  y0)  or  (x0  and  z0)  or  (y0  and  z0)))    (RJ ⁣(x,y,z,w)(0,))\left(x \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; z \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; w \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; \left(\left(x \ne 0 \;\mathbin{\operatorname{and}}\; y \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(x \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(y \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right)\right)\right) \implies \left(R_J\!\left(x, y, z, w\right) \in \left(0, \infty\right)\right)
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(x[0,)  and  y[0,)  and  z(0,)  and  (x0  or  y0))    (RD ⁣(x,y,z)(0,))\left(x \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; z \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; \left(x \ne 0 \;\mathbin{\operatorname{or}}\; y \ne 0\right)\right) \implies \left(R_D\!\left(x, y, z\right) \in \left(0, \infty\right)\right)
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(x[0,)  and  y(0,))    (RC ⁣(x,y)(0,))\left(x \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \left(0, \infty\right)\right) \implies \left(R_C\!\left(x, y\right) \in \left(0, \infty\right)\right)

Holomorphicity and branch cuts

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f(α) is holomorphic on αC(,0]   for all f{αRF ⁣(α,y,z),αRF ⁣(x,α,z),αRF ⁣(x,y,α)}f(\alpha) \text{ is holomorphic on } \alpha \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\text{ for all } f \in \left\{\alpha \mapsto R_F\!\left(\alpha, y, z\right), \alpha \mapsto R_F\!\left(x, \alpha, z\right), \alpha \mapsto R_F\!\left(x, y, \alpha\right)\right\}
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f(α) is holomorphic on αC(,0]   for all f{αRG ⁣(α,y,z),αRG ⁣(x,α,z),αRG ⁣(x,y,α)}f(\alpha) \text{ is holomorphic on } \alpha \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\text{ for all } f \in \left\{\alpha \mapsto R_G\!\left(\alpha, y, z\right), \alpha \mapsto R_G\!\left(x, \alpha, z\right), \alpha \mapsto R_G\!\left(x, y, \alpha\right)\right\}
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f(α) is holomorphic on αC(,0]   for all f{αRJ ⁣(α,y,z,w),αRJ ⁣(x,α,z,w),αRJ ⁣(x,y,α,w),αRJ ⁣(x,y,w,α)}f(\alpha) \text{ is holomorphic on } \alpha \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\text{ for all } f \in \left\{\alpha \mapsto R_J\!\left(\alpha, y, z, w\right), \alpha \mapsto R_J\!\left(x, \alpha, z, w\right), \alpha \mapsto R_J\!\left(x, y, \alpha, w\right), \alpha \mapsto R_J\!\left(x, y, w, \alpha\right)\right\}
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f(α) is holomorphic on αC(,0]   for all f{αRD ⁣(α,y,z),αRD ⁣(x,α,z),αRD ⁣(x,y,α)}f(\alpha) \text{ is holomorphic on } \alpha \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\text{ for all } f \in \left\{\alpha \mapsto R_D\!\left(\alpha, y, z\right), \alpha \mapsto R_D\!\left(x, \alpha, z\right), \alpha \mapsto R_D\!\left(x, y, \alpha\right)\right\}
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f(α) is holomorphic on αC(,0]   for all f{αRC ⁣(α,y),αRC ⁣(x,α)}f(\alpha) \text{ is holomorphic on } \alpha \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\text{ for all } f \in \left\{\alpha \mapsto R_C\!\left(\alpha, y\right), \alpha \mapsto R_C\!\left(x, \alpha\right)\right\}

Continuity on branch cuts

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RF ⁣(x,y,z)=limε0+RF ⁣(x+εi,y+εi,z+εi)R_F\!\left(x, y, z\right) = \lim_{\varepsilon \to {0}^{+}} R_F\!\left(x + \varepsilon i, y + \varepsilon i, z + \varepsilon i\right)
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RG ⁣(x,y,z)=limε0+RG ⁣(x+εi,y+εi,z+εi)R_G\!\left(x, y, z\right) = \lim_{\varepsilon \to {0}^{+}} R_G\!\left(x + \varepsilon i, y + \varepsilon i, z + \varepsilon i\right)
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RJ ⁣(x,y,z,w)=limε0+RJ ⁣(x+εi,y+εi,z+εi,w+εi)R_J\!\left(x, y, z, w\right) = \lim_{\varepsilon \to {0}^{+}} R_J\!\left(x + \varepsilon i, y + \varepsilon i, z + \varepsilon i, w + \varepsilon i\right)
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RD ⁣(x,y,z)=limε0+RD ⁣(x+εi,y+εi,z+εi)R_D\!\left(x, y, z\right) = \lim_{\varepsilon \to {0}^{+}} R_D\!\left(x + \varepsilon i, y + \varepsilon i, z + \varepsilon i\right)
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RC ⁣(x,y)=limε0+RC ⁣(x+εi,y+εi)R_C\!\left(x, y\right) = \lim_{\varepsilon \to {0}^{+}} R_C\!\left(x + \varepsilon i, y + \varepsilon i\right)

Representation of other functions

Related topics: Legendre elliptic integrals

Legendre complete elliptic integrals

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K(m)=RF ⁣(0,1m,1)K(m) = R_F\!\left(0, 1 - m, 1\right)
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E(m)=2RG ⁣(0,1m,1)E(m) = 2 R_G\!\left(0, 1 - m, 1\right)
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Π ⁣(n,m)=RF ⁣(0,1m,1)+n3RJ ⁣(0,1m,1,1n)\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)
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E(m)=1m3(RD ⁣(0,1m,1)+RD ⁣(0,1,1m))E(m) = \frac{1 - m}{3} \left(R_D\!\left(0, 1 - m, 1\right) + R_D\!\left(0, 1, 1 - m\right)\right)
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K(m)E(m)=m3RD ⁣(0,1m,1)K(m) - E(m) = \frac{m}{3} R_D\!\left(0, 1 - m, 1\right)
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E(m)(1m)K(m)=m(1m)3RD ⁣(0,1,1m)E(m) - \left(1 - m\right) K(m) = \frac{m \left(1 - m\right)}{3} R_D\!\left(0, 1, 1 - m\right)

Legendre incomplete elliptic integrals

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F ⁣(ϕ,m)=sin(ϕ)RF ⁣(cos2 ⁣(ϕ),1msin2 ⁣(ϕ),1)F\!\left(\phi, m\right) = \sin(\phi) R_F\!\left(\cos^{2}\!\left(\phi\right), 1 - m \sin^{2}\!\left(\phi\right), 1\right)
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E ⁣(ϕ,m)=sin(ϕ)RF ⁣(cos2 ⁣(ϕ),1msin2 ⁣(ϕ),1)13msin3 ⁣(ϕ)RD ⁣(cos2 ⁣(ϕ),1msin2 ⁣(ϕ),1)E\!\left(\phi, m\right) = \sin(\phi) R_F\!\left(\cos^{2}\!\left(\phi\right), 1 - m \sin^{2}\!\left(\phi\right), 1\right) - \frac{1}{3} m \sin^{3}\!\left(\phi\right) R_D\!\left(\cos^{2}\!\left(\phi\right), 1 - m \sin^{2}\!\left(\phi\right), 1\right)
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Π ⁣(n,ϕ,m)=sin(ϕ)RF ⁣(cos2 ⁣(ϕ),1msin2 ⁣(ϕ),1)+13nsin3 ⁣(ϕ)RJ ⁣(cos2 ⁣(ϕ),1msin2 ⁣(ϕ),1,1nsin2 ⁣(ϕ))\Pi\!\left(n, \phi, m\right) = \sin(\phi) R_F\!\left(\cos^{2}\!\left(\phi\right), 1 - m \sin^{2}\!\left(\phi\right), 1\right) + \frac{1}{3} n \sin^{3}\!\left(\phi\right) R_J\!\left(\cos^{2}\!\left(\phi\right), 1 - m \sin^{2}\!\left(\phi\right), 1, 1 - n \sin^{2}\!\left(\phi\right)\right)