# Fungrim entry: 618a9f

$\left|R_J\!\left(x, y, z, w\right) - {A}^{-3 / 2} \left(1 - \frac{3 E}{14} + \frac{F}{6} + \frac{9 {E}^{2}}{88} - \frac{3 G}{22} - \frac{9 E F}{52} + \frac{3 H}{26} - \frac{{E}^{3}}{16} + \frac{3 {F}^{2}}{40} + \frac{3 E G}{20} + \frac{45 {E}^{2} F}{272} - \frac{9 F G}{68} - \frac{9 E H}{68}\right)\right| \le \frac{3.4 \left|{A}^{-3 / 2}\right| {M}^{8}}{{\left(1 - M\right)}^{3 / 2}}\; \text{ where } A = \frac{x + y + z + 2 w}{5},\;X = 1 - \frac{x}{A},\;Y = 1 - \frac{y}{A},\;Z = 1 - \frac{z}{A},\;W = \frac{\left(-X - Y - Z\right)}{2},\;E = X Y + X Z + Y Z - 3 {W}^{2},\;F = X Y Z + 2 E W + 4 {W}^{3},\;G = \left(2 X Y Z + E W + 3 {W}^{3}\right) W,\;H = X Y Z {W}^{2},\;M = \max\!\left(\left|X\right|, \left|Y\right|, \left|Z\right|, \left|W\right|\right)$
Assumptions:$x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; w \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(x) \ge 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(y) \ge 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) \ge 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(w) > 0 \;\mathbin{\operatorname{and}}\; \left(\left(x \ne 0 \;\mathbin{\operatorname{and}}\; y \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(x \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(y \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right)\right) \;\mathbin{\operatorname{and}}\; \left|1 - \frac{5 x}{x + y + z + 2 w}\right| < 1 \;\mathbin{\operatorname{and}}\; \left|1 - \frac{5 y}{x + y + z + 2 w}\right| < 1 \;\mathbin{\operatorname{and}}\; \left|1 - \frac{5 z}{x + y + z + 2 w}\right| < 1 \;\mathbin{\operatorname{and}}\; \left|1 - \frac{5 z}{x + y + z + 2 w}\right| < 1$
References:
• https://doi.org/10.6028/jres.107.034
TeX:
\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)

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} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(x) \ge 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(y) \ge 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) \ge 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(w) > 0 \;\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}}\; \left|1 - \frac{5 x}{x + y + z + 2 w}\right| < 1 \;\mathbin{\operatorname{and}}\; \left|1 - \frac{5 y}{x + y + z + 2 w}\right| < 1 \;\mathbin{\operatorname{and}}\; \left|1 - \frac{5 z}{x + y + z + 2 w}\right| < 1 \;\mathbin{\operatorname{and}}\; \left|1 - \frac{5 z}{x + y + z + 2 w}\right| < 1
Definitions:
Fungrim symbol Notation Short description
Abs$\left|z\right|$ Absolute value
CarlsonRJ$R_J\!\left(x, y, z, w\right)$ Carlson symmetric elliptic integral of the third kind
Pow${a}^{b}$ Power
CC$\mathbb{C}$ Complex numbers
Re$\operatorname{Re}(z)$ Real part
Source code for this entry:
Entry(ID("618a9f"),
References("https://doi.org/10.6028/jres.107.034"))