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Fungrim entry: 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)
Assumptions: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))  and⁡  max⁡ ⁣(∣arg⁡(x)−arg⁡(y)∣,∣arg⁡(x)−arg⁡(z)∣,∣arg⁡(y)−arg⁡(z)∣)<π  and⁡  ∣1−3xx+y+z∣<1  and⁡  ∣1−3yx+y+z∣<1  and⁡  ∣1−3zx+y+z∣<1x \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) \;\mathbin{\operatorname{and}}\; \max\!\left(\left|\arg(x) - \arg(y)\right|, \left|\arg(x) - \arg(z)\right|, \left|\arg(y) - \arg(z)\right|\right) < \pi \;\mathbin{\operatorname{and}}\; \left|1 - \frac{3 x}{x + y + z}\right| < 1 \;\mathbin{\operatorname{and}}\; \left|1 - \frac{3 y}{x + y + z}\right| < 1 \;\mathbin{\operatorname{and}}\; \left|1 - \frac{3 z}{x + y + z}\right| < 1
References:
  • https://doi.org/10.6028/jres.107.034
TeX:
\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)

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) \;\mathbin{\operatorname{and}}\; \max\!\left(\left|\arg(x) - \arg(y)\right|, \left|\arg(x) - \arg(z)\right|, \left|\arg(y) - \arg(z)\right|\right) < \pi \;\mathbin{\operatorname{and}}\; \left|1 - \frac{3 x}{x + y + z}\right| < 1 \;\mathbin{\operatorname{and}}\; \left|1 - \frac{3 y}{x + y + z}\right| < 1 \;\mathbin{\operatorname{and}}\; \left|1 - \frac{3 z}{x + y + z}\right| < 1
Definitions:
Fungrim symbol Notation Short description
Abs∣z∣\left|z\right| Absolute value
CarlsonRFRF ⁣(x,y,z)R_F\!\left(x, y, z\right) Carlson symmetric elliptic integral of the first kind
Powab{a}^{b} Power
CCC\mathbb{C} Complex numbers
Argarg⁡(z)\arg(z) Complex argument
Piπ\pi The constant pi (3.14...)
Source code for this entry:
Entry(ID("799894"),
    Formula(Where(LessEqual(Abs(Sub(CarlsonRF(x, y, z), Mul(Pow(A, Neg(Div(1, 2))), Add(Add(Sub(Sub(Add(Add(Sub(1, Div(E, 10)), Div(F, 14)), Div(Pow(E, 2), 24)), Div(Mul(Mul(3, E), F), 44)), Div(Mul(5, Pow(E, 3)), 208)), Div(Mul(3, Pow(F, 2)), 104)), Div(Mul(Pow(E, 2), F), 16))))), Div(Mul(Mul(Decimal("0.2"), Abs(Pow(A, Neg(Div(1, 2))))), Pow(M, 8)), Sub(1, M))), Def(A, Div(Add(Add(x, y), z), 3)), Def(X, Sub(1, Div(x, A))), Def(Y, Sub(1, Div(y, A))), Def(Z, Sub(1, Div(z, A))), Def(E, Add(Add(Mul(X, Y), Mul(X, Z)), Mul(Y, Z))), Def(F, Mul(Mul(X, Y), Z)), Def(M, Max(Abs(X), Abs(Y), Abs(Z))))),
    Variables(x, y, z),
    Assumptions(And(Element(x, CC), Element(y, CC), Element(z, CC), Or(And(NotEqual(x, 0), NotEqual(y, 0)), And(NotEqual(x, 0), NotEqual(z, 0)), And(NotEqual(y, 0), NotEqual(z, 0))), Less(Max(Abs(Sub(Arg(x), Arg(y))), Abs(Sub(Arg(x), Arg(z))), Abs(Sub(Arg(y), Arg(z)))), Pi), Less(Abs(Sub(1, Div(Mul(3, x), Add(Add(x, y), z)))), 1), Less(Abs(Sub(1, Div(Mul(3, y), Add(Add(x, y), z)))), 1), Less(Abs(Sub(1, Div(Mul(3, z), Add(Add(x, y), z)))), 1))),
    References("https://doi.org/10.6028/jres.107.034"))

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2021-03-15 19:12:00.328586 UTC