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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)=12∫0∞1t+xt+yt+z dtR_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)=14∫0∞tt+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)=32∫0∞1(t+w)t+xt+yt+z dtR_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)=32∫0∞1(t+x)(t+y)(t+z)3/2 dtR_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)=12∫0∞1(t+y)t+x dtR_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/21ycos⁡2 ⁣(θ)+zsin⁡2 ⁣(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)=12∫0π/2ycos⁡2 ⁣(θ)+zsin⁡2 ⁣(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)=3∫0π/2sin⁡2 ⁣(θ)(ycos⁡2 ⁣(θ)+zsin⁡2 ⁣(θ))3/2 dθR_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)=14π∫02π∫0πsin⁡(θ)xsin⁡2 ⁣(θ)cos⁡2 ⁣(ϕ)+ysin⁡2 ⁣(θ)sin⁡2 ⁣(ϕ)+zcos⁡2 ⁣(θ) dθ dϕR_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)=14π∫02π∫0πxsin⁡2 ⁣(θ)cos⁡2 ⁣(ϕ)+ysin⁡2 ⁣(θ)sin⁡2 ⁣(ϕ)+zcos⁡2 ⁣(θ)sin⁡(θ) dθ dϕR_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)−(x−z)(y−z)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)=λ−1/2RF ⁣(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)=λ−3/2RJ ⁣(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)=λ−3/2RD ⁣(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)=λ−1/2RC ⁣(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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(x∈C  and⁡  y∈C  and⁡  z∈C  and⁡  ((x≠0  and⁡  y≠0)  or⁡  (x≠0  and⁡  z≠0)  or⁡  (y≠0  and⁡  z≠0)))    ⟹    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}
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(x∈C  and⁡  y∈C  and⁡  z∈C)    ⟹    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}
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(x∈C  and⁡  y∈C  and⁡  z∈C  and⁡  w∈C∖{0}  and⁡  ((x≠0  and⁡  y≠0)  or⁡  (x≠0  and⁡  z≠0)  or⁡  (y≠0  and⁡  z≠0)))    ⟹    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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(x∈C  and⁡  y∈C  and⁡  z∈C∖{0}  and⁡  (x≠0  or⁡  y≠0))    ⟹    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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(x∈C  and⁡  y∈C∖{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⁡  ((x≠0  and⁡  y≠0)  or⁡  (x≠0  and⁡  z≠0)  or⁡  (y≠0  and⁡  z≠0)))    ⟹    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⁡  ((x≠0  and⁡  y≠0)  or⁡  (x≠0  and⁡  z≠0)  or⁡  (y≠0  and⁡  z≠0)))    ⟹    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⁡  (x≠0  or⁡  y≠0))    ⟹    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 α∈C∖(−∞,0]   for all f∈{α↦RF ⁣(α,y,z),α↦RF ⁣(x,α,z),α↦RF ⁣(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 α∈C∖(−∞,0]   for all f∈{α↦RG ⁣(α,y,z),α↦RG ⁣(x,α,z),α↦RG ⁣(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 α∈C∖(−∞,0]   for all f∈{α↦RJ ⁣(α,y,z,w),α↦RJ ⁣(x,α,z,w),α↦RJ ⁣(x,y,α,w),α↦RJ ⁣(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 α∈C∖(−∞,0]   for all f∈{α↦RD ⁣(α,y,z),α↦RD ⁣(x,α,z),α↦RD ⁣(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 α∈C∖(−∞,0]   for all f∈{α↦RC ⁣(α,y),α↦RC ⁣(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)=lim⁡ε→0+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)=lim⁡ε→0+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)=lim⁡ε→0+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)=lim⁡ε→0+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)=lim⁡ε→0+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 ⁣(0,1−m,1)K(m) = R_F\!\left(0, 1 - m, 1\right)
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E(m)=2RG ⁣(0,1−m,1)E(m) = 2 R_G\!\left(0, 1 - m, 1\right)
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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)
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E(m)=1−m3(RD ⁣(0,1−m,1)+RD ⁣(0,1,1−m))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 ⁣(0,1−m,1)K(m) - E(m) = \frac{m}{3} R_D\!\left(0, 1 - m, 1\right)
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E(m)−(1−m)K(m)=m(1−m)3RD ⁣(0,1,1−m)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 ⁣(cos⁡2 ⁣(ϕ),1−msin⁡2 ⁣(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)
f48f54
E ⁣(ϕ,m)=sin⁡(ϕ)RF ⁣(cos⁡2 ⁣(ϕ),1−msin⁡2 ⁣(ϕ),1)−13msin⁡3 ⁣(ϕ)RD ⁣(cos⁡2 ⁣(ϕ),1−msin⁡2 ⁣(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 ⁣(cos⁡2 ⁣(ϕ),1−msin⁡2 ⁣(ϕ),1)+13nsin⁡3 ⁣(ϕ)RJ ⁣(cos⁡2 ⁣(ϕ),1−msin⁡2 ⁣(ϕ),1,1−nsin⁡2 ⁣(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)=(x−y)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,y2−x2)\operatorname{atanh}\!\left(\frac{x}{y}\right) = x R_C\!\left({y}^{2}, {y}^{2} - {x}^{2}\right)
584a61
asin⁡ ⁣(xy)=xRC ⁣(y2−x2,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)=y2−x2RC ⁣(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)=x2−y2RC ⁣(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 ⁣(z−e1 ⁣(τ),z−e2 ⁣(τ),z−e3 ⁣(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}
d38c27
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⁡ ⁣(yx−1)y−x,x≠y1x,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⁡ ⁣(xy−1)x−y,x≠y1x,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 ⁣(1−yx)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,x≠0y,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 ⁣(1−yx)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)=x−3/2R_D\!\left(x, x, x\right) = {x}^{-3 / 2}
8d0629
RD ⁣(0,y,z)=z−3/2{3(K ⁣(1−yz)−E ⁣(1−yz))1−yz,y≠z3π4,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(y−x)(RC ⁣(x,y)−xy),x≠yx−3/2,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)={3y−x(RC ⁣(y,x)−1y),x≠yx−3/2,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)=x−3/2R_J\!\left(x, x, x, x\right) = {x}^{-3 / 2}
d4b12e
RJ ⁣(x,y,y,w)={3w−y(RC ⁣(x,y)−RC ⁣(x,w)),y≠w32(y−x)(RC ⁣(x,y)−xy),y=w  and⁡  x≠yx−3/2,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=(w−x)(w−y)(w−z),  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

6dda7a
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
ddx RF ⁣(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
ddt RG ⁣(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
ddx RF ⁣(x,y,z)+ddy RF ⁣(x,y,z)+ddz RF ⁣(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
ddx RG ⁣(x,y,z)+ddy RG ⁣(x,y,z)+ddz RG ⁣(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
xddx RF ⁣(x,y,z)+yddy RF ⁣(x,y,z)+zddz RF ⁣(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
ddx RC ⁣(x,y)={12(y−x)(RC ⁣(x,y)−1x),x≠y−16x−3/2,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
ddy RC ⁣(x,y)={12(x−y)(RC ⁣(x,y)−xy),x≠y−13x−3/2,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 ⁣(1,12,32,1−x)R_C\!\left(1, x\right) = \,{}_2F_1\!\left(1, \frac{1}{2}, \frac{3}{2}, 1 - x\right)
b2fdfe
RF ⁣(0,x,1)=π2 2F1 ⁣(12,12,1,1−x)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)=π4 2F1 ⁣(−12,12,1,1−x)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 — R−a ⁣(b,z)R_{-a}\!\left(b, z\right) — Carlson multivariate hypergeometric function
8f71cb
RF ⁣(x,y,z)=R−1/2 ⁣([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)=R−3/2 ⁣([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)−A−1/2(1−E10+F14+E224−3EF44−5E3208+3F2104+E2F16)∣≤0.2∣A−1/2∣M81−M   where A=x+y+z3,  X=1−xA,  Y=1−yA,  Z=1−zA,  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