# Carlson symmetric elliptic integrals

## Definitions

Symbol: CarlsonRF $R_F\!\left(x, y, z\right)$ Carlson symmetric elliptic integral of the first kind
Symbol: CarlsonRG $R_G\!\left(x, y, z\right)$ Carlson symmetric elliptic integral of the second kind
Symbol: CarlsonRJ $R_J\!\left(x, y, z, w\right)$ Carlson symmetric elliptic integral of the third kind
Symbol: CarlsonRD $R_D\!\left(x, y, z\right)$ Degenerate Carlson symmetric elliptic integral of the third kind
Symbol: CarlsonRC $R_C\!\left(x, y\right)$ Degenerate Carlson symmetric elliptic integral of the first kind

## Illustrations

Image: Plot of $R_F\!\left(x, y, 1\right)$ on $x \in \left[0, 4\right]$ for $y \in \left\{0.1, 1, 10\right\}$
Image: Plot of $R_G\!\left(x, y, 1\right)$ on $x \in \left[0, 4\right]$ for $y \in \left\{0.1, 1, 10\right\}$
Image: X-ray of $R_F\!\left(z, 1, 2\right)$ on $z \in \left[-4, 4\right] + \left[-4, 4\right] i$

## Integral representations

### Defining algebraic integrals

$R_F\!\left(x, y, z\right) = \frac{1}{2} \int_{0}^{\infty} \frac{1}{\sqrt{t + x} \sqrt{t + y} \sqrt{t + z}} \, dt$
$R_G\!\left(x, y, z\right) = \frac{1}{4} \int_{0}^{\infty} \frac{t}{\sqrt{t + x} \sqrt{t + y} \sqrt{t + z}} \left(\frac{x}{t + x} + \frac{y}{t + y} + \frac{z}{t + z}\right) \, dt$
$R_J\!\left(x, y, z, w\right) = \frac{3}{2} \int_{0}^{\infty} \frac{1}{\left(t + w\right) \sqrt{t + x} \sqrt{t + y} \sqrt{t + z}} \, dt$
$R_D\!\left(x, y, z\right) = \frac{3}{2} \int_{0}^{\infty} \frac{1}{\left(t + x\right) \left(t + y\right) {\left(t + z\right)}^{3 / 2}} \, dt$
$R_C\!\left(x, y\right) = \frac{1}{2} \int_{0}^{\infty} \frac{1}{\left(t + y\right) \sqrt{t + x}} \, dt$

### Trigonometric integrals

$R_F\!\left(0, y, z\right) = \int_{0}^{\pi / 2} \frac{1}{\sqrt{y \cos^{2}\!\left(\theta\right) + z \sin^{2}\!\left(\theta\right)}} \, d\theta$
$R_G\!\left(0, y, z\right) = \frac{1}{2} \int_{0}^{\pi / 2} \sqrt{y \cos^{2}\!\left(\theta\right) + z \sin^{2}\!\left(\theta\right)} \, d\theta$
$R_D\!\left(0, y, z\right) = 3 \int_{0}^{\pi / 2} \frac{\sin^{2}\!\left(\theta\right)}{{\left(y \cos^{2}\!\left(\theta\right) + z \sin^{2}\!\left(\theta\right)\right)}^{3 / 2}} \, d\theta$
$R_F\!\left(x, y, z\right) = \frac{1}{4 \pi} \int_{0}^{2 \pi} \int_{0}^{\pi} \frac{\sin(\theta)}{\sqrt{x \sin^{2}\!\left(\theta\right) \cos^{2}\!\left(\phi\right) + y \sin^{2}\!\left(\theta\right) \sin^{2}\!\left(\phi\right) + z \cos^{2}\!\left(\theta\right)}} \, d\theta \, d\phi$
$R_G\!\left(x, y, z\right) = \frac{1}{4 \pi} \int_{0}^{2 \pi} \int_{0}^{\pi} \sqrt{x \sin^{2}\!\left(\theta\right) \cos^{2}\!\left(\phi\right) + y \sin^{2}\!\left(\theta\right) \sin^{2}\!\left(\phi\right) + z \cos^{2}\!\left(\theta\right)} \sin(\theta) \, d\theta \, d\phi$

## Connection formulas

### Degenerate cases

$R_C\!\left(x, y\right) = R_F\!\left(x, y, y\right)$
$R_D\!\left(x, y, z\right) = R_J\!\left(x, y, z, z\right)$

### Connection formula for RF, RG and RD

$2 R_G\!\left(x, y, z\right) = z R_F\!\left(x, y, z\right) - \frac{\left(x - z\right) \left(y - z\right)}{3} R_D\!\left(x, y, z\right) + \frac{\sqrt{x} \sqrt{y}}{\sqrt{z}}$

## Symmetry

$R_F\!\left(x, y, z\right) = R_F\!\left(x, z, y\right) = R_F\!\left(y, x, z\right) = R_F\!\left(y, z, x\right) = R_F\!\left(z, x, y\right) = R_F\!\left(z, y, x\right)$
$R_G\!\left(x, y, z\right) = R_G\!\left(x, z, y\right) = R_G\!\left(y, x, z\right) = R_G\!\left(y, z, x\right) = R_G\!\left(z, x, y\right) = R_G\!\left(z, y, x\right)$
$R_J\!\left(x, y, z, w\right) = R_J\!\left(x, z, y, w\right) = R_J\!\left(y, x, z, w\right) = R_J\!\left(y, z, x, w\right) = R_J\!\left(z, x, y, w\right) = R_J\!\left(z, y, x, w\right)$
$R_D\!\left(x, y, z\right) = R_D\!\left(y, x, z\right)$

## Scale invariance

$R_F\!\left(\lambda x, \lambda y, \lambda z\right) = {\lambda}^{-1 / 2} R_F\!\left(x, y, z\right)$
$R_G\!\left(\lambda x, \lambda y, \lambda z\right) = {\lambda}^{1 / 2} R_G\!\left(x, y, z\right)$
$R_J\!\left(\lambda x, \lambda y, \lambda z, \lambda w\right) = {\lambda}^{-3 / 2} R_J\!\left(x, y, z, w\right)$
$R_D\!\left(\lambda x, \lambda y, \lambda z\right) = {\lambda}^{-3 / 2} R_D\!\left(x, y, z\right)$
$R_C\!\left(\lambda x, \lambda y\right) = {\lambda}^{-1 / 2} R_C\!\left(x, y\right)$

## Domain

### Complex variables

$\left(x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(\left(x \ne 0 \;\mathbin{\operatorname{and}}\; y \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(x \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(y \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right)\right)\right) \;\implies\; R_F\!\left(x, y, z\right) \in \mathbb{C}$
$\left(x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}\right) \;\implies\; R_G\!\left(x, y, z\right) \in \mathbb{C}$
$\left(x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; w \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; \left(\left(x \ne 0 \;\mathbin{\operatorname{and}}\; y \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(x \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(y \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right)\right)\right) \;\implies\; R_J\!\left(x, y, z, w\right) \in \mathbb{C}$
$\left(x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; \left(x \ne 0 \;\mathbin{\operatorname{or}}\; y \ne 0\right)\right) \;\implies\; R_D\!\left(x, y, z\right) \in \mathbb{C}$
$\left(x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \setminus \left\{0\right\}\right) \;\implies\; R_C\!\left(x, y\right) \in \mathbb{C}$

### Real variables

$\left(x \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; z \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; \left(\left(x \ne 0 \;\mathbin{\operatorname{and}}\; y \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(x \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(y \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right)\right)\right) \;\implies\; R_F\!\left(x, y, z\right) \in \left(0, \infty\right)$
$\left(x \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; z \in \left[0, \infty\right)\right) \;\implies\; R_G\!\left(x, y, z\right) \in \left[0, \infty\right)$
$\left(x \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; z \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; w \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; \left(\left(x \ne 0 \;\mathbin{\operatorname{and}}\; y \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(x \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(y \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right)\right)\right) \;\implies\; R_J\!\left(x, y, z, w\right) \in \left(0, \infty\right)$
$\left(x \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; z \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; \left(x \ne 0 \;\mathbin{\operatorname{or}}\; y \ne 0\right)\right) \;\implies\; R_D\!\left(x, y, z\right) \in \left(0, \infty\right)$
$\left(x \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \left(0, \infty\right)\right) \;\implies\; R_C\!\left(x, y\right) \in \left(0, \infty\right)$

### Holomorphicity and branch cuts

$f(\alpha) \text{ is holomorphic on } \alpha \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\text{ for all } f \in \left\{\alpha \mapsto R_F\!\left(\alpha, y, z\right), \alpha \mapsto R_F\!\left(x, \alpha, z\right), \alpha \mapsto R_F\!\left(x, y, \alpha\right)\right\}$
$f(\alpha) \text{ is holomorphic on } \alpha \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\text{ for all } f \in \left\{\alpha \mapsto R_G\!\left(\alpha, y, z\right), \alpha \mapsto R_G\!\left(x, \alpha, z\right), \alpha \mapsto R_G\!\left(x, y, \alpha\right)\right\}$
$f(\alpha) \text{ is holomorphic on } \alpha \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\text{ for all } f \in \left\{\alpha \mapsto R_J\!\left(\alpha, y, z, w\right), \alpha \mapsto R_J\!\left(x, \alpha, z, w\right), \alpha \mapsto R_J\!\left(x, y, \alpha, w\right), \alpha \mapsto R_J\!\left(x, y, w, \alpha\right)\right\}$
$f(\alpha) \text{ is holomorphic on } \alpha \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\text{ for all } f \in \left\{\alpha \mapsto R_D\!\left(\alpha, y, z\right), \alpha \mapsto R_D\!\left(x, \alpha, z\right), \alpha \mapsto R_D\!\left(x, y, \alpha\right)\right\}$
$f(\alpha) \text{ is holomorphic on } \alpha \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\text{ for all } f \in \left\{\alpha \mapsto R_C\!\left(\alpha, y\right), \alpha \mapsto R_C\!\left(x, \alpha\right)\right\}$

### Continuity on branch cuts

$R_F\!\left(x, y, z\right) = \lim_{\varepsilon \to {0}^{+}} R_F\!\left(x + \varepsilon i, y + \varepsilon i, z + \varepsilon i\right)$
$R_G\!\left(x, y, z\right) = \lim_{\varepsilon \to {0}^{+}} R_G\!\left(x + \varepsilon i, y + \varepsilon i, z + \varepsilon i\right)$
$R_J\!\left(x, y, z, w\right) = \lim_{\varepsilon \to {0}^{+}} R_J\!\left(x + \varepsilon i, y + \varepsilon i, z + \varepsilon i, w + \varepsilon i\right)$
$R_D\!\left(x, y, z\right) = \lim_{\varepsilon \to {0}^{+}} R_D\!\left(x + \varepsilon i, y + \varepsilon i, z + \varepsilon i\right)$
$R_C\!\left(x, y\right) = \lim_{\varepsilon \to {0}^{+}} R_C\!\left(x + \varepsilon i, y + \varepsilon i\right)$

## Representation of other functions

Related topics: Legendre elliptic integrals

### Legendre complete elliptic integrals

$K(m) = R_F\!\left(0, 1 - m, 1\right)$
$E(m) = 2 R_G\!\left(0, 1 - m, 1\right)$
$\Pi\!\left(n, m\right) = R_F\!\left(0, 1 - m, 1\right) + \frac{n}{3} R_J\!\left(0, 1 - m, 1, 1 - n\right)$
$E(m) = \frac{1 - m}{3} \left(R_D\!\left(0, 1 - m, 1\right) + R_D\!\left(0, 1, 1 - m\right)\right)$
$K(m) - E(m) = \frac{m}{3} R_D\!\left(0, 1 - m, 1\right)$
$E(m) - \left(1 - m\right) K(m) = \frac{m \left(1 - m\right)}{3} R_D\!\left(0, 1, 1 - m\right)$

### Legendre incomplete elliptic integrals

$F\!\left(\phi, m\right) = \sin(\phi) R_F\!\left(\cos^{2}\!\left(\phi\right), 1 - m \sin^{2}\!\left(\phi\right), 1\right)$
$E\!\left(\phi, m\right) = \sin(\phi) R_F\!\left(\cos^{2}\!\left(\phi\right), 1 - m \sin^{2}\!\left(\phi\right), 1\right) - \frac{1}{3} m \sin^{3}\!\left(\phi\right) R_D\!\left(\cos^{2}\!\left(\phi\right), 1 - m \sin^{2}\!\left(\phi\right), 1\right)$
$\Pi\!\left(n, \phi, m\right) = \sin(\phi) R_F\!\left(\cos^{2}\!\left(\phi\right), 1 - m \sin^{2}\!\left(\phi\right), 1\right) + \frac{1}{3} n \sin^{3}\!\left(\phi\right) R_J\!\left(\cos^{2}\!\left(\phi\right), 1 - m \sin^{2}\!\left(\phi\right), 1, 1 - n \sin^{2}\!\left(\phi\right)\right)$