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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
dab889
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
02a8d7
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
0d8639
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)
b478a1
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)
655a2b
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\; R_F\!\left(x, y, z\right) \in \mathbb{C}
c90834
(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\; R_G\!\left(x, y, z\right) \in \mathbb{C}
8bac89
(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\; R_J\!\left(x, y, z, w\right) \in \mathbb{C}
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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\; R_D\!\left(x, y, z\right) \in \mathbb{C}
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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\; R_C\!\left(x, y\right) \in \mathbb{C}

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\; R_F\!\left(x, y, z\right) \in \left(0, \infty\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\; R_G\!\left(x, y, z\right) \in \left[0, \infty\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\; R_J\!\left(x, y, z, w\right) \in \left(0, \infty\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\; R_D\!\left(x, y, z\right) \in \left(0, \infty\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\; R_C\!\left(x, y\right) \in \left(0, \infty\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\}
114f9e
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)
9673f7
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)
55d23d
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)
8f4e31
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)

Elementary functions

398bb7
log ⁣(xy)=(xy)RC ⁣((x+y)24,xy)\log\!\left(\frac{x}{y}\right) = \left(x - y\right) R_C\!\left(\frac{{\left(x + y\right)}^{2}}{4}, x y\right)
7a9dad
atan ⁣(xy)=xRC ⁣(y2,y2+x2)\operatorname{atan}\!\left(\frac{x}{y}\right) = x R_C\!\left({y}^{2}, {y}^{2} + {x}^{2}\right)
2cdd2f
atanh ⁣(xy)=xRC ⁣(y2,y2x2)\operatorname{atanh}\!\left(\frac{x}{y}\right) = x R_C\!\left({y}^{2}, {y}^{2} - {x}^{2}\right)
584a61
asin ⁣(xy)=xRC ⁣(y2x2,y2)\operatorname{asin}\!\left(\frac{x}{y}\right) = x R_C\!\left({y}^{2} - {x}^{2}, {y}^{2}\right)
423b36
asinh ⁣(xy)=xRC ⁣(y2+x2,y2)\operatorname{asinh}\!\left(\frac{x}{y}\right) = x R_C\!\left({y}^{2} + {x}^{2}, {y}^{2}\right)
33e034
acos ⁣(xy)=y2x2RC ⁣(x2,y2)\operatorname{acos}\!\left(\frac{x}{y}\right) = \sqrt{{y}^{2} - {x}^{2}} R_C\!\left({x}^{2}, {y}^{2}\right)
d9765b
acosh ⁣(xy)=x2y2RC ⁣(x2,y2)\operatorname{acosh}\!\left(\frac{x}{y}\right) = \sqrt{{x}^{2} - {y}^{2}} R_C\!\left({x}^{2}, {y}^{2}\right)

Inverse Weierstrass elliptic function

Related topics: Weierstrass elliptic functions

124339
 ⁣(f(z),τ)=z   where f(z)=RF ⁣(ze1 ⁣(τ),ze2 ⁣(τ),ze3 ⁣(τ))\wp\!\left(f(z), \tau\right) = z\; \text{ where } f(z) = R_F\!\left(z - e_{1}\!\left(\tau\right), z - e_{2}\!\left(\tau\right), z - e_{3}\!\left(\tau\right)\right)

Specific values

Related topics: Specific values of Carlson symmetric elliptic integrals

The degenerate integral of the first kind

e464ec
RC ⁣(0,1)=π2R_C\!\left(0, 1\right) = \frac{\pi}{2}
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RC ⁣(1,1)=1R_C\!\left(1, 1\right) = 1
eac389
RC ⁣(1,2)=π4R_C\!\left(1, 2\right) = \frac{\pi}{4}
a15c03
RC ⁣(2,1)=log ⁣(1+2)R_C\!\left(2, 1\right) = \log\!\left(1 + \sqrt{2}\right)
7b5755
RC ⁣(x,y)={atan ⁣(yx1)yx,xy1x,x=yR_C\!\left(x, y\right) = \begin{cases} \frac{\operatorname{atan}\!\left(\sqrt{\frac{y}{x} - 1}\right)}{\sqrt{y - x}}, & x \ne y\\\frac{1}{\sqrt{x}}, & x = y\\ \end{cases}

The integral of the first kind

8bb972
RF ⁣(0,1,1)=π2R_F\!\left(0, 1, 1\right) = \frac{\pi}{2}
c166ca
RF ⁣(1,1,1)=1R_F\!\left(1, 1, 1\right) = 1
4cd504
RF ⁣(1,1,2)=log ⁣(1+2)R_F\!\left(1, 1, 2\right) = \log\!\left(1 + \sqrt{2}\right)
0bf328
RF ⁣(1,2,2)=π4R_F\!\left(1, 2, 2\right) = \frac{\pi}{4}
28237a
RF ⁣(0,1,2)=(Γ ⁣(14))242πR_F\!\left(0, 1, 2\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}}
9b0388
RF ⁣(x,x,x)=1xR_F\!\left(x, x, x\right) = \frac{1}{\sqrt{x}}
ebaa1a
RF ⁣(x,x,y)=RC ⁣(y,x)R_F\!\left(x, x, y\right) = R_C\!\left(y, x\right)
649dc0
RF ⁣(x,x,y)={atan ⁣(xy1)xy,xy1x,x=yR_F\!\left(x, x, y\right) = \begin{cases} \frac{\operatorname{atan}\!\left(\sqrt{\frac{x}{y} - 1}\right)}{\sqrt{x - y}}, & x \ne y\\\frac{1}{\sqrt{x}}, & x = y\\ \end{cases}
415ff0
RF ⁣(0,x,y)=K ⁣(1yx)xR_F\!\left(0, x, y\right) = \frac{K\!\left(1 - \frac{y}{x}\right)}{\sqrt{x}}

The integral of the second kind

bcc121
RG ⁣(0,0,0)=0R_G\!\left(0, 0, 0\right) = 0
d5ff09
RG ⁣(0,0,1)=12R_G\!\left(0, 0, 1\right) = \frac{1}{2}
cd55cf
RG ⁣(0,1,1)=π4R_G\!\left(0, 1, 1\right) = \frac{\pi}{4}
250ff1
RG ⁣(1,1,1)=1R_G\!\left(1, 1, 1\right) = 1
4d7098
RG ⁣(1,1,2)=22+log ⁣(1+2)2R_G\!\left(1, 1, 2\right) = \frac{\sqrt{2}}{2} + \frac{\log\!\left(1 + \sqrt{2}\right)}{2}
d51efc
RG ⁣(1,2,2)=π4+12R_G\!\left(1, 2, 2\right) = \frac{\pi}{4} + \frac{1}{2}
84f403
RG ⁣(0,1,2)=(Γ ⁣(14))282π+π3/22(Γ ⁣(14))2R_G\!\left(0, 1, 2\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{8 \sqrt{2 \pi}} + \frac{{\pi}^{3 / 2}}{\sqrt{2} {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}
990145
RG ⁣(x,x,x)=xR_G\!\left(x, x, x\right) = \sqrt{x}
d829be
RG ⁣(0,0,x)=x2R_G\!\left(0, 0, x\right) = \frac{\sqrt{x}}{2}
120284
RG ⁣(x,x,y)=12{xRC ⁣(y,x)+y,x0y,x=0R_G\!\left(x, x, y\right) = \frac{1}{2} \begin{cases} x R_C\!\left(y, x\right) + \sqrt{y}, & x \ne 0\\\sqrt{y}, & x = 0\\ \end{cases}
7cddc6
RG ⁣(0,x,y)=xE ⁣(1yx)2R_G\!\left(0, x, y\right) = \frac{\sqrt{x} E\!\left(1 - \frac{y}{x}\right)}{2}

The degenerate integral of the third kind

1c0fee
RD ⁣(1,1,1)=1R_D\!\left(1, 1, 1\right) = 1
84ea08
RD ⁣(0,1,1)=3π4R_D\!\left(0, 1, 1\right) = \frac{3 \pi}{4}
63644d
RD ⁣(0,2,1)=32π3/2(Γ ⁣(14))2R_D\!\left(0, 2, 1\right) = \frac{3 \sqrt{2} {\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}
f47947
RD ⁣(1,1,2)=3log ⁣(1+2)322R_D\!\left(1, 1, 2\right) = 3 \log\!\left(1 + \sqrt{2}\right) - \frac{3 \sqrt{2}}{2}
ccb4d1
RD ⁣(x,x,x)=x3/2R_D\!\left(x, x, x\right) = {x}^{-3 / 2}
8d0629
RD ⁣(0,y,z)=z3/2{3(K ⁣(1yz)E ⁣(1yz))1yz,yz3π4,y=zR_D\!\left(0, y, z\right) = {z}^{-3 / 2} \begin{cases} \frac{3 \left(K\!\left(1 - \frac{y}{z}\right) - E\!\left(1 - \frac{y}{z}\right)\right)}{1 - \frac{y}{z}}, & y \ne z\\\frac{3 \pi}{4}, & y = z\\ \end{cases}
c85c2f
RD ⁣(x,y,y)={32(yx)(RC ⁣(x,y)xy),xyx3/2,x=yR_D\!\left(x, y, y\right) = \begin{cases} \frac{3}{2 \left(y - x\right)} \left(R_C\!\left(x, y\right) - \frac{\sqrt{x}}{y}\right), & x \ne y\\{x}^{-3 / 2}, & x = y\\ \end{cases}
771801
RD ⁣(x,x,y)={3yx(RC ⁣(y,x)1y),xyx3/2,x=yR_D\!\left(x, x, y\right) = \begin{cases} \frac{3}{y - x} \left(R_C\!\left(y, x\right) - \frac{1}{\sqrt{y}}\right), & x \ne y\\{x}^{-3 / 2}, & x = y\\ \end{cases}

The integral of the third kind

e9d5a9
RJ ⁣(1,1,1,1)=1R_J\!\left(1, 1, 1, 1\right) = 1
6e9544
RJ ⁣(1,1,2,4)=log ⁣(1+2)2π8R_J\!\left(1, 1, 2, 4\right) = \log\!\left(1 + \sqrt{2}\right) - \frac{\sqrt{2} \pi}{8}
1eaaed
RJ ⁣(0,i,i,1)=3(Γ ⁣(14))28πR_J\!\left(0, i, -i, 1\right) = \frac{3 {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{8 \sqrt{\pi}}
4c882a
RJ ⁣(x,x,x,x)=x3/2R_J\!\left(x, x, x, x\right) = {x}^{-3 / 2}
d4b12e
RJ ⁣(x,y,y,w)={3wy(RC ⁣(x,y)RC ⁣(x,w)),yw32(yx)(RC ⁣(x,y)xy),y=w  and  xyx3/2,x=y=wR_J\!\left(x, y, y, w\right) = \begin{cases} \frac{3}{w - y} \left(R_C\!\left(x, y\right) - R_C\!\left(x, w\right)\right), & y \ne w\\\frac{3}{2 \left(y - x\right)} \left(R_C\!\left(x, y\right) - \frac{\sqrt{x}}{y}\right), & y = w \;\mathbin{\operatorname{and}}\; x \ne y\\{x}^{-3 / 2}, & x = y = w\\ \end{cases}
f6b4a2
RJ ⁣(0,x,x,w)=3π2(xw+wx)R_J\!\left(0, x, x, w\right) = \frac{3 \pi}{2 \left(x \sqrt{w} + w \sqrt{x}\right)}
3dd30a
RJ ⁣(x,y,z,z)=RD ⁣(x,y,z)R_J\!\left(x, y, z, z\right) = R_D\!\left(x, y, z\right)
5c6f10
RJ ⁣(x,x,x,w)=RD ⁣(w,w,x)R_J\!\left(x, x, x, w\right) = R_D\!\left(w, w, x\right)

Functional equations

Duplication theorems

2499cd
RF ⁣(x,y,z)=2RF ⁣(x+λ,y+λ,z+λ)   where λ=xy+yz+xzR_F\!\left(x, y, z\right) = 2 R_F\!\left(x + \lambda, y + \lambda, z + \lambda\right)\; \text{ where } \lambda = \sqrt{x} \sqrt{y} + \sqrt{y} \sqrt{z} + \sqrt{x} \sqrt{z}
8e6189
RF ⁣(x,y,z)=RF ⁣(x+λ4,y+λ4,z+λ4)   where λ=xy+yz+xzR_F\!\left(x, y, z\right) = R_F\!\left(\frac{x + \lambda}{4}, \frac{y + \lambda}{4}, \frac{z + \lambda}{4}\right)\; \text{ where } \lambda = \sqrt{x} \sqrt{y} + \sqrt{y} \sqrt{z} + \sqrt{x} \sqrt{z}
47cf5d
RG ⁣(x,y,z)=2RG ⁣(x+λ,y+λ,z+λ)12(λRF ⁣(x,y,z)+x+y+z)   where λ=xy+yz+xzR_G\!\left(x, y, z\right) = 2 R_G\!\left(x + \lambda, y + \lambda, z + \lambda\right) - \frac{1}{2} \left(\lambda R_F\!\left(x, y, z\right) + \sqrt{x} + \sqrt{y} + \sqrt{z}\right)\; \text{ where } \lambda = \sqrt{x} \sqrt{y} + \sqrt{y} \sqrt{z} + \sqrt{x} \sqrt{z}
31a3ba
RD ⁣(x,y,z)=2RD ⁣(x+λ,y+λ,z+λ)+3z(z+λ)   where λ=xy+yz+xzR_D\!\left(x, y, z\right) = 2 R_D\!\left(x + \lambda, y + \lambda, z + \lambda\right) + \frac{3}{\sqrt{z} \left(z + \lambda\right)}\; \text{ where } \lambda = \sqrt{x} \sqrt{y} + \sqrt{y} \sqrt{z} + \sqrt{x} \sqrt{z}
791c44
RJ ⁣(x,y,z,w)=2RJ ⁣(x+λ,y+λ,z+λ,w+λ)+6dRC ⁣(1,1+δd2)   where λ=xy+yz+xz,  δ=(wx)(wy)(wz),  d=(w+x)(w+y)(w+z)R_J\!\left(x, y, z, w\right) = 2 R_J\!\left(x + \lambda, y + \lambda, z + \lambda, w + \lambda\right) + \frac{6}{d} R_C\!\left(1, 1 + \frac{\delta}{{d}^{2}}\right)\; \text{ where } \lambda = \sqrt{x} \sqrt{y} + \sqrt{y} \sqrt{z} + \sqrt{x} \sqrt{z},\;\delta = \left(w - x\right) \left(w - y\right) \left(w - z\right),\;d = \left(\sqrt{w} + \sqrt{x}\right) \left(\sqrt{w} + \sqrt{y}\right) \left(\sqrt{w} + \sqrt{z}\right)
8f5d76
RC ⁣(x,y)=2RC ⁣(x+λ,y+λ)   where λ=y+2xyR_C\!\left(x, y\right) = 2 R_C\!\left(x + \lambda, y + \lambda\right)\; \text{ where } \lambda = y + 2 \sqrt{x} \sqrt{y}

Addition theorems

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RD ⁣(x,y,z)+RD ⁣(y,z,x)+RD ⁣(z,x,y)=3xyzR_D\!\left(x, y, z\right) + R_D\!\left(y, z, x\right) + R_D\!\left(z, x, y\right) = \frac{3}{\sqrt{x} \sqrt{y} \sqrt{z}}
38fa65
RF ⁣(x+λ,y+λ,λ)+RF ⁣(x+μ,y+μ,μ)=RF ⁣(x,y,0)   where μ=xyλR_F\!\left(x + \lambda, y + \lambda, \lambda\right) + R_F\!\left(x + \mu, y + \mu, \mu\right) = R_F\!\left(x, y, 0\right)\; \text{ where } \mu = \frac{x y}{\lambda}
4eac3f
RJ ⁣(x+λ,y+λ,λ,w+λ)+RJ ⁣(x+μ,y+μ,μ,w+μ)=RJ ⁣(x,y,0,w)3RC ⁣(w2(λ+μ+x+y),w(w+λ)(w+μ))   where μ=xyλR_J\!\left(x + \lambda, y + \lambda, \lambda, w + \lambda\right) + R_J\!\left(x + \mu, y + \mu, \mu, w + \mu\right) = R_J\!\left(x, y, 0, w\right) - 3 R_C\!\left({w}^{2} \left(\lambda + \mu + x + y\right), w \left(w + \lambda\right) \left(w + \mu\right)\right)\; \text{ where } \mu = \frac{x y}{\lambda}
a203e9
RD ⁣(λ,x+λ,y+λ)+RD ⁣(μ,x+μ,y+μ)=RD ⁣(0,x,y)3yx+y+λ+μ   where μ=xyλR_D\!\left(\lambda, x + \lambda, y + \lambda\right) + R_D\!\left(\mu, x + \mu, y + \mu\right) = R_D\!\left(0, x, y\right) - \frac{3}{y \sqrt{x + y + \lambda + \mu}}\; \text{ where } \mu = \frac{x y}{\lambda}

Derivatives and differential equations

5f0adb
ddxRF ⁣(x,y,z)=16RD ⁣(y,z,x)\frac{d}{d x}\, R_F\!\left(x, y, z\right) = -\frac{1}{6} R_D\!\left(y, z, x\right)
638fa6
ddtRG ⁣(x+t,y+t,z+t)=12RF ⁣(x+t,y+t,z+t)\frac{d}{d t}\, R_G\!\left(x + t, y + t, z + t\right) = \frac{1}{2} R_F\!\left(x + t, y + t, z + t\right)
ce327b
ddxRF ⁣(x,y,z)+ddyRF ⁣(x,y,z)+ddzRF ⁣(x,y,z)=1xyz\frac{d}{d x}\, R_F\!\left(x, y, z\right) + \frac{d}{d y}\, R_F\!\left(x, y, z\right) + \frac{d}{d z}\, R_F\!\left(x, y, z\right) = -\frac{1}{\sqrt{x} \sqrt{y} \sqrt{z}}
3e1435
ddxRG ⁣(x,y,z)+ddyRG ⁣(x,y,z)+ddzRG ⁣(x,y,z)=12RF ⁣(x,y,z)\frac{d}{d x}\, R_G\!\left(x, y, z\right) + \frac{d}{d y}\, R_G\!\left(x, y, z\right) + \frac{d}{d z}\, R_G\!\left(x, y, z\right) = \frac{1}{2} R_F\!\left(x, y, z\right)
644d75
xddxRF ⁣(x,y,z)+yddyRF ⁣(x,y,z)+zddzRF ⁣(x,y,z)=12RF ⁣(x,y,z)x \frac{d}{d x}\, R_F\!\left(x, y, z\right) + y \frac{d}{d y}\, R_F\!\left(x, y, z\right) + z \frac{d}{d z}\, R_F\!\left(x, y, z\right) = -\frac{1}{2} R_F\!\left(x, y, z\right)
de8485
ddxRC ⁣(x,y)={12(yx)(RC ⁣(x,y)1x),xy16x3/2,x=y\frac{d}{d x}\, R_C\!\left(x, y\right) = \begin{cases} \frac{1}{2 \left(y - x\right)} \left(R_C\!\left(x, y\right) - \frac{1}{\sqrt{x}}\right), & x \ne y\\-\frac{1}{6} {x}^{-3 / 2}, & x = y\\ \end{cases}
741859
ddyRC ⁣(x,y)={12(xy)(RC ⁣(x,y)xy),xy13x3/2,x=y\frac{d}{d y}\, R_C\!\left(x, y\right) = \begin{cases} \frac{1}{2 \left(x - y\right)} \left(R_C\!\left(x, y\right) - \frac{\sqrt{x}}{y}\right), & x \ne y\\-\frac{1}{3} {x}^{-3 / 2}, & x = y\\ \end{cases}

Series representations

Related topics: Series representations of Carlson symmetric elliptic integrals

Cases reducible to the Gauss hypergeometric function

72b5bd
RC ⁣(1,x)=2F1 ⁣(1,12,32,1x)R_C\!\left(1, x\right) = \,{}_2F_1\!\left(1, \frac{1}{2}, \frac{3}{2}, 1 - x\right)
b2fdfe
RF ⁣(0,x,1)=π22F1 ⁣(12,12,1,1x)R_F\!\left(0, x, 1\right) = \frac{\pi}{2} \,{}_2F_1\!\left(\frac{1}{2}, \frac{1}{2}, 1, 1 - x\right)
e98dd0
RG ⁣(0,x,1)=π42F1 ⁣(12,12,1,1x)R_G\!\left(0, x, 1\right) = \frac{\pi}{4} \,{}_2F_1\!\left(-\frac{1}{2}, \frac{1}{2}, 1, 1 - x\right)

Multivariate hypergeometric series representations

b576e6
Symbol: CarlsonHypergeometricR Ra ⁣(b,z)R_{-a}\!\left(b, z\right) Carlson multivariate hypergeometric function
8f71cb
RF ⁣(x,y,z)=R1/2 ⁣([12,12,12],[x,y,z])R_F\!\left(x, y, z\right) = R_{-1 / 2}\!\left(\left[\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right], \left[x, y, z\right]\right)
fda084
RG ⁣(x,y,z)=R1/2 ⁣([12,12,12],[x,y,z])R_G\!\left(x, y, z\right) = R_{1 / 2}\!\left(\left[\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right], \left[x, y, z\right]\right)
b2cd79
RJ ⁣(x,y,z,w)=R3/2 ⁣([12,12,12,1],[x,y,z,w])R_J\!\left(x, y, z, w\right) = R_{-3 / 2}\!\left(\left[\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, 1\right], \left[x, y, z, w\right]\right)

Explicit truncated series approximations

799894
RF ⁣(x,y,z)A1/2(1E10+F14+E2243EF445E3208+3F2104+E2F16)0.2A1/2M81M   where A=x+y+z3,  X=1xA,  Y=1yA,  Z=1zA,  E=XY+XZ+YZ,  F=XYZ,  M=max ⁣(X,Y,Z)\left|R_F\!\left(x, y, z\right) - {A}^{-1 / 2} \left(1 - \frac{E}{10} + \frac{F}{14} + \frac{{E}^{2}}{24} - \frac{3 E F}{44} - \frac{5 {E}^{3}}{208} + \frac{3 {F}^{2}}{104} + \frac{{E}^{2} F}{16}\right)\right| \le \frac{0.2 \left|{A}^{-1 / 2}\right| {M}^{8}}{1 - M}\; \text{ where } A = \frac{x + y + z}{3},\;X = 1 - \frac{x}{A},\;Y = 1 - \frac{y}{A},\;Z = 1 - \frac{z}{A},\;E = X Y + X Z + Y Z,\;F = X Y Z,\;M = \max\!\left(\left|X\right|, \left|Y\right|, \left|Z\right|\right)

Bounds and inequalities

Real variables

c03f78
RF ⁣(x,y,z)1(xyz)1/6R_F\!\left(x, y, z\right) \le \frac{1}{{\left(x y z\right)}^{1 / 6}}
1d2811
RF ⁣(x,y,z)3x+y+zR_F\!\left(x, y, z\right) \ge \frac{3}{\sqrt{x} + \sqrt{y} + \sqrt{z}}
07584a
RG ⁣(x,y,z)min ⁣(x+y+z3,x2+y2+z23xyz)R_G\!\left(x, y, z\right) \le \min\!\left(\sqrt{\frac{x + y + z}{3}}, \frac{{x}^{2} + {y}^{2} + {z}^{2}}{3 \sqrt{x y z}}\right)
edcf6c
RG ⁣(x,y,z)x+y+z3R_G\!\left(x, y, z\right) \ge \frac{\sqrt{x} + \sqrt{y} + \sqrt{z}}{3}
a4e47f
RJ ⁣(x,y,z,w)(xyzw2)3/10R_J\!\left(x, y, z, w\right) \le {\left(x y z {w}^{2}\right)}^{-3 / 10}
d3b39c
RJ ⁣(x,y,z,w)(5x+y+z+2w)3R_J\!\left(x, y, z, w\right) \ge {\left(\frac{5}{\sqrt{x} + \sqrt{y} + \sqrt{z} + 2 \sqrt{w}}\right)}^{3}
230a49
RD ⁣(x,y,z)(xyz3)3/10R_D\!\left(x, y, z\right) \le {\left(x y {z}^{3}\right)}^{-3 / 10}
34e932
RD ⁣(x,y,z)(5x+y+3z)3R_D\!\left(x, y, z\right) \ge {\left(\frac{5}{\sqrt{x} + \sqrt{y} + 3 \sqrt{z}}\right)}^{3}
688efb
RC ⁣(x,y)1(xy2)1/6R_C\!\left(x, y\right) \le \frac{1}{{\left(x {y}^{2}\right)}^{1 / 6}}
978287
RC ⁣(x,y)3x+2yR_C\!\left(x, y\right) \ge \frac{3}{\sqrt{x} + 2 \sqrt{y}}

Real variables, complete integrals

0e209c
RF ⁣(0,y,z)12max ⁣(y,z)(π+log ⁣(yz))R_F\!\left(0, y, z\right) \le \frac{1}{2 \sqrt{\max\!\left(y, z\right)}} \left(\pi + \left|\log\!\left(\frac{y}{z}\right)\right|\right)
2e40b8
RF ⁣(0,y,z)2log(2)max ⁣(y,z)R_F\!\left(0, y, z\right) \ge \frac{2 \log(2)}{\sqrt{\max\!\left(y, z\right)}}
62eade
RG ⁣(0,y,z)πmax ⁣(y,z)4R_G\!\left(0, y, z\right) \le \frac{\pi \sqrt{\max\!\left(y, z\right)}}{4}
36ae10
RG ⁣(0,y,z)max ⁣(y,z)2R_G\!\left(0, y, z\right) \ge \frac{\sqrt{\max\!\left(y, z\right)}}{2}
add3ea
RJ ⁣(0,y,z,w)3π4(yzw2)3/8R_J\!\left(0, y, z, w\right) \le \frac{3 \pi}{4} {\left(y z {w}^{2}\right)}^{-3 / 8}
60541a
RJ ⁣(0,y,z,w)3π2w(2yz+yw+zw)R_J\!\left(0, y, z, w\right) \ge \frac{3 \pi}{2 \sqrt{w \left(2 y z + y w + z w\right)}}
d70b12
RD ⁣(0,y,z)3π4(yz3)3/8R_D\!\left(0, y, z\right) \le \frac{3 \pi}{4} {\left(y {z}^{3}\right)}^{-3 / 8}
255142
RD ⁣(0,y,z)3π2z3y+y+zR_D\!\left(0, y, z\right) \ge \frac{3 \pi}{2 z \sqrt{3 y + y + z}}

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