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Series representations of Carlson symmetric elliptic integrals

Table of contents: Definitions - Cases reducible to the Gauss hypergeometric function - Incomplete integrals - Complete integrals - General formulas for the series - Symmetric formulas - Approximations by truncated series - Integral representations - Gauss hypergeometric series

Related topics: Carlson symmetric elliptic integrals

Definitions

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Symbol: CarlsonHypergeometricR — R−a ⁣(b,z)R_{-a}\!\left(b, z\right) — Carlson multivariate hypergeometric function
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Symbol: CarlsonHypergeometricT — TN ⁣(b,z)T_{N}\!\left(b, z\right) — Term in expansion of Carlson multivariate hypergeometric function

Cases reducible to the Gauss hypergeometric function

Related topics: Gauss hypergeometric function

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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)
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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)
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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)
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RD ⁣(0,x,1)=3π4 2F1 ⁣(12,32,2,1−x)R_D\!\left(0, x, 1\right) = \frac{3 \pi}{4} \,{}_2F_1\!\left(\frac{1}{2}, \frac{3}{2}, 2, 1 - x\right)
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RD ⁣(0,1,x)=3π4x 2F1 ⁣(12,12,2,1−x)R_D\!\left(0, 1, x\right) = \frac{3 \pi}{4 x} \,{}_2F_1\!\left(\frac{1}{2}, \frac{1}{2}, 2, 1 - x\right)

Incomplete integrals

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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)
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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)
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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)
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RJ ⁣(x,y,z,w)=R−3/2 ⁣([12,12,12,12,12],[x,y,z,w,w])R_J\!\left(x, y, z, w\right) = R_{-3 / 2}\!\left(\left[\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right], \left[x, y, z, w, w\right]\right)
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RD ⁣(x,y,z)=R−3/2 ⁣([12,12,32],[x,y,z])R_D\!\left(x, y, z\right) = R_{-3 / 2}\!\left(\left[\frac{1}{2}, \frac{1}{2}, \frac{3}{2}\right], \left[x, y, z\right]\right)
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RD ⁣(x,y,z)=R−3/2 ⁣([12,12,12,12,12],[x,y,z,z,z])R_D\!\left(x, y, z\right) = R_{-3 / 2}\!\left(\left[\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right], \left[x, y, z, z, z\right]\right)
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RC ⁣(x,y)=R−1/2 ⁣([12,1],[x,y])R_C\!\left(x, y\right) = R_{-1 / 2}\!\left(\left[\frac{1}{2}, 1\right], \left[x, y\right]\right)
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RC ⁣(x,y)=R−1/2 ⁣([12,12,12],[x,y,y])R_C\!\left(x, y\right) = R_{-1 / 2}\!\left(\left[\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right], \left[x, y, y\right]\right)

Complete integrals

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R−a ⁣([b1,b2,…,bn],[0,z2,z3,…,zn])=B ⁣(a,c−b1)B ⁣(a,c)R−a ⁣([b2,b3,…,bn],[z2,z3,…,zn])   where c=−a+∑k=1nbkR_{-a}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\right], \left[0, z_{2}, z_{3}, \ldots, z_{n}\right]\right) = \frac{\mathrm{B}\!\left(a, c - b_{1}\right)}{\mathrm{B}\!\left(a, c\right)} R_{-a}\!\left(\left[b_{2}, b_{3}, \ldots, b_{n}\right], \left[z_{2}, z_{3}, \ldots, z_{n}\right]\right)\; \text{ where } c = -a + \sum_{k=1}^{n} b_{k}
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RF ⁣(0,y,z)=π2R−1/2 ⁣([12,12],[y,z])R_F\!\left(0, y, z\right) = \frac{\pi}{2} R_{-1 / 2}\!\left(\left[\frac{1}{2}, \frac{1}{2}\right], \left[y, z\right]\right)
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RD ⁣(0,y,z)=3π4R−3/2 ⁣([12,32],[y,z])R_D\!\left(0, y, z\right) = \frac{3 \pi}{4} R_{-3 / 2}\!\left(\left[\frac{1}{2}, \frac{3}{2}\right], \left[y, z\right]\right)
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RD ⁣(0,y,z)=3π4R−3/2 ⁣([12,12,12,12],[y,z,z,z])R_D\!\left(0, y, z\right) = \frac{3 \pi}{4} R_{-3 / 2}\!\left(\left[\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right], \left[y, z, z, z\right]\right)
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RJ ⁣(0,y,z,w)=3π4R−3/2 ⁣([12,12,1],[y,z,w])R_J\!\left(0, y, z, w\right) = \frac{3 \pi}{4} R_{-3 / 2}\!\left(\left[\frac{1}{2}, \frac{1}{2}, 1\right], \left[y, z, w\right]\right)
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RJ ⁣(0,y,z,w)=3π4R−3/2 ⁣([12,12,12,12],[y,z,w,w])R_J\!\left(0, y, z, w\right) = \frac{3 \pi}{4} R_{-3 / 2}\!\left(\left[\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right], \left[y, z, w, w\right]\right)
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RG ⁣(0,y,z)=π4R1/2 ⁣([12,12],[y,z])R_G\!\left(0, y, z\right) = \frac{\pi}{4} R_{1 / 2}\!\left(\left[\frac{1}{2}, \frac{1}{2}\right], \left[y, z\right]\right)

General formulas for the series

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TN ⁣([b1,b2,…,bn],[z1,z2,…,zn])=∑(m1,m2,…,mn)∈(Z≥0)n∑k=1nmk=N∏k=1n(bk)mk(mk)!zkmkT_{N}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right) = \sum_{\textstyle{\left(m_{1}, m_{2}, \ldots, m_{n}\right) \in {\left(\mathbb{Z}_{\ge 0}\right)}^{n} \atop \sum_{k=1}^{n} m_{k} = N}} \prod_{k=1}^{n} \frac{\left(b_{k}\right)_{m_{k}}}{\left(m_{k}\right)!} z_{k}^{m_{k}}
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R−a ⁣([b1,b2,…,bn],[z1,z2,…,zn])=∑N=0∞(a)N(c)NTN ⁣([b1,b2,…,bn],[1−z1,1−z2,…,1−zn])   where c=∑k=1nbkR_{-a}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right) = \sum_{N=0}^{\infty} \frac{\left(a\right)_{N}}{\left(c\right)_{N}} T_{N}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\right], \left[1 - z_{1}, 1 - z_{2}, \ldots, 1 - z_{n}\right]\right)\; \text{ where } c = \sum_{k=1}^{n} b_{k}
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R−a ⁣([b1,b2,…,bn],[z1,z2,…,zn])=zn−a∑N=0∞(a)N(c)NTN ⁣([b1,b2,…,bn−1],[1−z1zn,1−z2zn,…,1−zn−1zn])   where c=∑k=1nbkR_{-a}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right) = z_{n}^{-a} \sum_{N=0}^{\infty} \frac{\left(a\right)_{N}}{\left(c\right)_{N}} T_{N}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n - 1}\right], \left[1 - \frac{z_{1}}{z_{n}}, 1 - \frac{z_{2}}{z_{n}}, \ldots, 1 - \frac{z_{n - 1}}{z_{n}}\right]\right)\; \text{ where } c = \sum_{k=1}^{n} b_{k}

Symmetric formulas

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R−a ⁣([β,…,β⏟n times],[z1,z2,…,zn])=A−aR−a ⁣([β,…,β⏟n times],[z1A,z2A,…,znA])   where A=1n∑k=1nzkR_{-a}\!\left(\left[\underbrace{\beta, \ldots, \beta}_{n \text{ times}}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right) = {A}^{-a} R_{-a}\!\left(\left[\underbrace{\beta, \ldots, \beta}_{n \text{ times}}\right], \left[\frac{z_{1}}{A}, \frac{z_{2}}{A}, \ldots, \frac{z_{n}}{A}\right]\right)\; \text{ where } A = \frac{1}{n} \sum_{k=1}^{n} z_{k}
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R−a ⁣([β,…,β⏟n times],[z1,z2,…,zn])=A−a∑N=0∞(a)N(nβ)NTN ⁣([β,…,β⏟n times],[1−z1A,1−z2A,…,1−znA])   where A=1n∑k=1nzkR_{-a}\!\left(\left[\underbrace{\beta, \ldots, \beta}_{n \text{ times}}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right) = {A}^{-a} \sum_{N=0}^{\infty} \frac{\left(a\right)_{N}}{\left(n \beta\right)_{N}} T_{N}\!\left(\left[\underbrace{\beta, \ldots, \beta}_{n \text{ times}}\right], \left[1 - \frac{z_{1}}{A}, 1 - \frac{z_{2}}{A}, \ldots, 1 - \frac{z_{n}}{A}\right]\right)\; \text{ where } A = \frac{1}{n} \sum_{k=1}^{n} z_{k}
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TN ⁣([β,…,β⏟n times],[z1,z2,…,zn])=∑(m1,m2,…,mn)∈(Z≥0)n∑k=1nkmk=N(−1)M+N(β)M∏k=1nekmk ⁣([z1,z2,…,zn])(mk)!   where M=∑k=1nmkT_{N}\!\left(\left[\underbrace{\beta, \ldots, \beta}_{n \text{ times}}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right) = \sum_{\textstyle{\left(m_{1}, m_{2}, \ldots, m_{n}\right) \in {\left(\mathbb{Z}_{\ge 0}\right)}^{n} \atop \sum_{k=1}^{n} k m_{k} = N}} {\left(-1\right)}^{M + N} \left(\beta\right)_{M} \prod_{k=1}^{n} \frac{e_{k}^{m_{k}}\!\left(\left[z_{1}, z_{2}, \ldots, z_{n}\right]\right)}{\left(m_{k}\right)!}\; \text{ where } M = \sum_{k=1}^{n} m_{k}
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R−a ⁣([b1,b2,…,bn],[λz1,λz2,…,λzn])=λ−aR−a ⁣([b1,b2,…,bn],[z1,z2,…,zn])R_{-a}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\right], \left[\lambda z_{1}, \lambda z_{2}, \ldots, \lambda z_{n}\right]\right) = {\lambda}^{-a} R_{-a}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right)
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R−a ⁣([b1,b2,…,bn],[z,…,z⏟n times])=z−aR_{-a}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\right], \left[\underbrace{z, \ldots, z}_{n \text{ times}}\right]\right) = {z}^{-a}

Approximations by truncated series

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∣R−a ⁣([β,…,β⏟n times],[z1,z2,…,zn])−A−a∑N=0K−1(a)N(nβ)NTN ⁣([β,…,β⏟n times],[z1,z2,…,zn])∣≤∣A−a∣(∣a∣)KMKK!(1−M)max⁡(∣a∣,1)   where A=1n∑k=1nzk,  Zk=1−zkA,  M=max⁡ ⁣(∣Z1∣,∣Z2∣,…,∣Zn∣)\left|R_{-a}\!\left(\left[\underbrace{\beta, \ldots, \beta}_{n \text{ times}}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right) - {A}^{-a} \sum_{N=0}^{K - 1} \frac{\left(a\right)_{N}}{\left(n \beta\right)_{N}} T_{N}\!\left(\left[\underbrace{\beta, \ldots, \beta}_{n \text{ times}}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right)\right| \le \frac{\left|{A}^{-a}\right| \left(\left|a\right|\right)_{K} {M}^{K}}{K ! {\left(1 - M\right)}^{\max\left(\left|a\right|, 1\right)}}\; \text{ where } A = \frac{1}{n} \sum_{k=1}^{n} z_{k},\;Z_{k} = 1 - \frac{z_{k}}{A},\;M = \max\!\left(\left|Z_{1}\right|, \left|Z_{2}\right|, \ldots, \left|Z_{n}\right|\right)
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∣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)
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∣RJ ⁣(x,y,z,w)−A−3/2(1−3E14+F6+9E288−3G22−9EF52+3H26−E316+3F240+3EG20+45E2F272−9FG68−9EH68)∣≤3.4∣A−3/2∣M8(1−M)3/2   where A=x+y+z+2w5,  X=1−xA,  Y=1−yA,  Z=1−zA,  W=(−X−Y−Z)2,  E=XY+XZ+YZ−3W2,  F=XYZ+2EW+4W3,  G=(2XYZ+EW+3W3)W,  H=XYZW2,  M=max⁡ ⁣(∣X∣,∣Y∣,∣Z∣,∣W∣)\left|R_J\!\left(x, y, z, w\right) - {A}^{-3 / 2} \left(1 - \frac{3 E}{14} + \frac{F}{6} + \frac{9 {E}^{2}}{88} - \frac{3 G}{22} - \frac{9 E F}{52} + \frac{3 H}{26} - \frac{{E}^{3}}{16} + \frac{3 {F}^{2}}{40} + \frac{3 E G}{20} + \frac{45 {E}^{2} F}{272} - \frac{9 F G}{68} - \frac{9 E H}{68}\right)\right| \le \frac{3.4 \left|{A}^{-3 / 2}\right| {M}^{8}}{{\left(1 - M\right)}^{3 / 2}}\; \text{ where } A = \frac{x + y + z + 2 w}{5},\;X = 1 - \frac{x}{A},\;Y = 1 - \frac{y}{A},\;Z = 1 - \frac{z}{A},\;W = \frac{\left(-X - Y - Z\right)}{2},\;E = X Y + X Z + Y Z - 3 {W}^{2},\;F = X Y Z + 2 E W + 4 {W}^{3},\;G = \left(2 X Y Z + E W + 3 {W}^{3}\right) W,\;H = X Y Z {W}^{2},\;M = \max\!\left(\left|X\right|, \left|Y\right|, \left|Z\right|, \left|W\right|\right)

Integral representations

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R−a ⁣([b1,b2,…,bn],[z1,z2,…,zn])=1B ⁣(a,c)∫0∞tc−1∏k=1n(t+zk)−bk dt   where c=−a+∑j=1nbjR_{-a}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right) = \frac{1}{\mathrm{B}\!\left(a, c\right)} \int_{0}^{\infty} {t}^{c - 1} \prod_{k=1}^{n} {\left(t + z_{k}\right)}^{-b_{k}} \, dt\; \text{ where } c = -a + \sum_{j=1}^{n} b_{j}
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R−a ⁣([b1,b2,…,bn],[z1,z2,…,zn])=1B ⁣(a,c)∫0∞ta−1∏k=1n(1+tzk)−bk dt   where c=−a+∑j=1nbjR_{-a}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right) = \frac{1}{\mathrm{B}\!\left(a, c\right)} \int_{0}^{\infty} {t}^{a - 1} \prod_{k=1}^{n} {\left(1 + t z_{k}\right)}^{-b_{k}} \, dt\; \text{ where } c = -a + \sum_{j=1}^{n} b_{j}

Gauss hypergeometric series

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RC ⁣(1,1+y)= 2F1 ⁣(1,12,32,−y)R_C\!\left(1, 1 + y\right) = \,{}_2F_1\!\left(1, \frac{1}{2}, \frac{3}{2}, -y\right)

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2021-03-15 19:12:00.328586 UTC