# Specific values of Carlson symmetric elliptic integrals

Related topics: Carlson symmetric elliptic integrals

## The elementary integral RC

### Scale invariance

$R_C\!\left(\lambda x, \lambda y\right) = {\lambda}^{-1 / 2} R_C\!\left(x, y\right)$

### Particular constant values

$R_C\!\left(0, 0\right) = {\tilde \infty}$
$R_C\!\left(1, 0\right) = \infty$
$R_C\!\left(0, 1\right) = \frac{\pi}{2}$
$R_C\!\left(1, 1\right) = 1$
$R_C\!\left(1, 2\right) = \frac{\pi}{4}$
$R_C\!\left(2, 1\right) = \log\!\left(1 + \sqrt{2}\right)$
$R_C\!\left(0, -1\right) = -\frac{\pi i}{2}$
$R_C\!\left(-1, 0\right) = -i \infty$
$R_C\!\left(1, -1\right) = \frac{\sqrt{2} \log\!\left(1 + \sqrt{2}\right)}{2} - \frac{\pi \sqrt{2}}{4} i$
$R_C\!\left(-1, 1\right) = \frac{\pi \sqrt{2}}{4} - \frac{\sqrt{2} \log\!\left(1 + \sqrt{2}\right)}{2} i$

### Specialized values

$R_C\!\left(x, 0\right) = \begin{cases} \operatorname{sgn}\!\left(\frac{1}{\sqrt{x}}\right) \infty, & x \ne 0\\{\tilde \infty}, & x = 0\\ \end{cases}$
$R_C\!\left(0, y\right) = \begin{cases} \frac{\pi}{2 \sqrt{y}}, & y \ne 0\\{\tilde \infty}, & y = 0\\ \end{cases}$
$R_C\!\left(x, x\right) = \frac{1}{\sqrt{x}}$
$R_C\!\left(x, 2 x\right) = \frac{\pi}{4 \sqrt{x}}$
$R_C\!\left(2 x, x\right) = \frac{\log\!\left(1 + \sqrt{2}\right)}{\sqrt{x}}$

### General formulas for real variables

$R_C\!\left(x, y\right) = \begin{cases} \frac{\operatorname{atan}\!\left(\sqrt{\frac{y}{x} - 1}\right)}{\sqrt{y - x}}, & x < y\\\frac{1}{\sqrt{x}}, & x = y\\\frac{\operatorname{atanh}\!\left(\sqrt{1 - \frac{y}{x}}\right)}{\sqrt{x - y}}, & x > y\\ \end{cases}$
$R_C\!\left(x, y\right) = \begin{cases} \frac{\operatorname{acos}\!\left(\sqrt{\frac{x}{y}}\right)}{\sqrt{y - x}}, & x < y\\\frac{1}{\sqrt{x}}, & x = y\\\frac{\operatorname{acosh}\!\left(\sqrt{\frac{x}{y}}\right)}{\sqrt{x - y}}, & x > y\\ \end{cases}$
$R_C\!\left(-x, -y\right) = -i R_C\!\left(x, y\right)$
$R_C\!\left(x, -y\right) = \frac{1}{\sqrt{x + y}} \left(\operatorname{atanh}\!\left(\sqrt{\frac{x}{x + y}}\right) - \frac{\pi i}{2}\right)$
$R_C\!\left(-x, y\right) = \frac{1}{\sqrt{x + y}} \left(\frac{\pi}{2} - \operatorname{atanh}\!\left(\sqrt{\frac{x}{x + y}}\right) i\right)$
$R_C\!\left(-x, y\right) = \overline{i R_C\!\left(x, -y\right)}$

### General formulas for one or more complex variables

$R_C\!\left(x, y\right) = \begin{cases} \frac{\operatorname{atan}\!\left(\sqrt{\frac{y}{x} - 1}\right)}{\sqrt{y - x}}, & x \ne y\\\frac{1}{\sqrt{x}}, & x = y\\ \end{cases}$
$R_C\!\left(x, y\right) = \begin{cases} \frac{\operatorname{atanh}\!\left(\sqrt{1 - \frac{y}{x}}\right)}{\sqrt{x - y}}, & x \ne y\\\frac{1}{\sqrt{x}}, & x = y\\ \end{cases}$
$R_C\!\left(x, c x\right) = \begin{cases} \frac{\operatorname{atan}\!\left(\sqrt{c - 1}\right)}{\sqrt{\left(c - 1\right) x}}, & c > 1\\\frac{1}{\sqrt{x}}, & c = 1\\\frac{\operatorname{atanh}\!\left(\sqrt{1 - c}\right)}{\sqrt{\left(1 - c\right) x}}, & c < 1\\ \end{cases}$
$R_C\!\left(x, -c x\right) = \frac{1}{\sqrt{\left(c + 1\right) x}} \begin{cases} \operatorname{atanh}\!\left(\sqrt{c + 1}\right), & \operatorname{Im}(x) < 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(x) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(x) \ge 0\right)\\\operatorname{atanh}\!\left(\sqrt{c + 1}\right) + \pi i, & \text{otherwise}\\ \end{cases}$
$R_C\!\left(1, 1 + y\right) = \begin{cases} \frac{\operatorname{atan}\!\left(\sqrt{y}\right)}{\sqrt{y}}, & y \ne 0\\1, & y = 0\\ \end{cases}$
$R_C\!\left(1, 1 + y\right) = \,{}_2F_1\!\left(1, \frac{1}{2}, \frac{3}{2}, -y\right)$

## The integral of the first kind RF

### Symmetry and scale invariance

$R_F\!\left(x, y, z\right) = R_F\!\left(x, z, y\right) = R_F\!\left(y, x, z\right) = R_F\!\left(y, z, x\right) = R_F\!\left(z, x, y\right) = R_F\!\left(z, y, x\right)$
$R_F\!\left(\lambda x, \lambda y, \lambda z\right) = {\lambda}^{-1 / 2} R_F\!\left(x, y, z\right)$

### Particular constant values

$R_F\!\left(0, 0, 0\right) = {\tilde \infty}$
$R_F\!\left(0, 0, 1\right) = \infty$
$R_F\!\left(0, 1, 1\right) = \frac{\pi}{2}$
$R_F\!\left(1, 1, 1\right) = 1$
$R_F\!\left(1, 1, 2\right) = \log\!\left(1 + \sqrt{2}\right)$
$R_F\!\left(1, 2, 2\right) = \frac{\pi}{4}$
$R_F\!\left(0, 1, 2\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}}$
$R_F\!\left(0, 1, -1\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}} \left(1 - i\right)$
$R_F\!\left(0, 2, 4\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{8 \sqrt{\pi}}$
$R_F\!\left(0, \frac{1}{2}, 1\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{\pi}}$
$R_F\!\left(0, 0, -1\right) = -i \infty$
$R_F\!\left(0, -1, -1\right) = \frac{-\pi i}{2}$
$R_F\!\left(-1, -1, -1\right) = -i$
$R_F\!\left(0, -1, -2\right) = -\frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}} i$
$R_F\!\left(0, 1, 12 \sqrt{2} - 16\right) = \frac{\left(2 + \sqrt{2}\right) {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{16 \sqrt{\pi}}$
$R_F\!\left(0, i, -i\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{\pi}}$
$R_F\!\left(0, \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{4}}{16 \pi}, \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{4}}{32 \pi}\right) = 1$
$R_F\!\left(0, \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{4}}{32 \pi}, \frac{-{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{4}}{32 \pi}\right) = 1 - i$

### Special parametric cases

$R_F\!\left(0, 0, x\right) = \begin{cases} \infty, & x \ne 0\\{\tilde \infty}, & x = 0\\ \end{cases}$
$R_F\!\left(0, 1, x\right) = K\!\left(1 - x\right)$
$R_F\!\left(0, x, x\right) = \frac{\pi}{2 \sqrt{x}}$
$R_F\!\left(0, x, y\right) = \frac{K\!\left(1 - \frac{y}{x}\right)}{\sqrt{x}}$
$R_F\!\left(0, x, 2 x\right) = \frac{1}{\sqrt{x}} \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}}$
$R_F\!\left(0, x, -x\right) = \frac{1}{\sqrt{x}} \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}} \begin{cases} 1 - i, & \operatorname{Im}(x) < 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(x) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(x) \ge 0\right)\\1 + i, & \text{otherwise}\\ \end{cases}$
$R_F\!\left(0, x, c x\right) = \frac{K\!\left(1 - c\right)}{\sqrt{x}}$
$R_F\!\left(0, x, -c x\right) = \frac{1}{\sqrt{x}} \begin{cases} K\!\left(1 + c\right), & \operatorname{Im}(x) < 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(x) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(x) \ge 0\right)\\K\!\left(1 + c\right) + 2 i K\!\left(-c\right), & \text{otherwise}\\ \end{cases}$
$R_F\!\left(x, y, y\right) = R_C\!\left(x, y\right)$
$R_F\!\left(x, x, y\right) = R_C\!\left(y, x\right)$
$R_F\!\left(x, x, y\right) = \begin{cases} \frac{\operatorname{atan}\!\left(\sqrt{\frac{x}{y} - 1}\right)}{\sqrt{x - y}}, & x \ne y\\\frac{1}{\sqrt{x}}, & x = y\\ \end{cases}$
$R_F\!\left(x, x, x\right) = \frac{1}{\sqrt{x}}$
$R_F\!\left(-x, -y, -z\right) = -i R_F\!\left(x, y, z\right)$
$R_F\!\left(-x, -y, z\right) = \overline{i R_F\!\left(x, y, -z\right)}$

## The integral of the second kind RG

### Symmetry and scale invariance

$R_G\!\left(x, y, z\right) = R_G\!\left(x, z, y\right) = R_G\!\left(y, x, z\right) = R_G\!\left(y, z, x\right) = R_G\!\left(z, x, y\right) = R_G\!\left(z, y, x\right)$
$R_G\!\left(\lambda x, \lambda y, \lambda z\right) = {\lambda}^{1 / 2} R_G\!\left(x, y, z\right)$

### Particular constant values

$R_G\!\left(0, 0, 0\right) = 0$
$R_G\!\left(0, 0, 1\right) = \frac{1}{2}$
$R_G\!\left(0, 1, 1\right) = \frac{\pi}{4}$
$R_G\!\left(1, 1, 1\right) = 1$
$R_G\!\left(1, 1, 2\right) = \frac{\sqrt{2}}{2} + \frac{\log\!\left(1 + \sqrt{2}\right)}{2}$
$R_G\!\left(1, 2, 2\right) = \frac{\pi}{4} + \frac{1}{2}$
$R_G\!\left(0, 1, 2\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{8 \sqrt{2 \pi}} + \frac{{\pi}^{3 / 2}}{\sqrt{2} {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}$
$R_G\!\left(0, 1, -1\right) = \frac{\sqrt{2} {\pi}^{3 / 2}}{2 {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}} \left(1 + i\right)$
$R_G\!\left(0, 16, 16\right) = \pi$

### Special parametric cases

$R_G\!\left(0, 0, x\right) = \frac{\sqrt{x}}{2}$
$R_G\!\left(0, 1, x\right) = \frac{E\!\left(1 - x\right)}{2}$
$R_G\!\left(0, x, x\right) = \frac{\pi \sqrt{x}}{4}$
$R_G\!\left(0, x, y\right) = \frac{\sqrt{x} E\!\left(1 - \frac{y}{x}\right)}{2}$
$R_G\!\left(0, x, 2 x\right) = \sqrt{x} \left(\frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{8 \sqrt{2 \pi}} + \frac{{\pi}^{3 / 2}}{\sqrt{2} {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}\right)$
$R_G\!\left(0, x, -x\right) = \sqrt{x} \frac{\sqrt{2} {\pi}^{3 / 2}}{2 {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}} \begin{cases} 1 + i, & \operatorname{Im}(x) < 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(x) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(x) \ge 0\right)\\1 - i, & \text{otherwise}\\ \end{cases}$