# Legendre elliptic integrals

## Definitions

Symbol: EllipticK $K(m)$ Legendre complete elliptic integral of the first kind
Symbol: EllipticE $E(m)$ Legendre complete elliptic integral of the second kind
Symbol: EllipticPi $\Pi\!\left(n, m\right)$ Legendre complete elliptic integral of the third kind
Symbol: IncompleteEllipticF $F\!\left(\phi, m\right)$ Legendre incomplete elliptic integral of the first kind
Symbol: IncompleteEllipticE $E\!\left(\phi, m\right)$ Legendre incomplete elliptic integral of the second kind
Symbol: IncompleteEllipticPi $\Pi\!\left(n, \phi, m\right)$ Legendre incomplete elliptic integral of the third kind

## Illustrations

Image: Plot of $K(m)$ on $m \in \left[-2, 2\right]$
Image: Plot of $E(m)$ on $m \in \left[-2, 2\right]$
Image: Plot of $F\!\left(\phi, m\right)$ on $\phi \in \left[-2 \pi, 2 \pi\right]$
Image: Plot of $E\!\left(\phi, m\right)$ on $\phi \in \left[-2 \pi, 2 \pi\right]$

## Integral representations

### Trigonometric forms of the complete integrals

$K(m) = \int_{0}^{\pi / 2} \frac{1}{\sqrt{1 - m \sin^{2}\!\left(x\right)}} \, dx$
$E(m) = \int_{0}^{\pi / 2} \sqrt{1 - m \sin^{2}\!\left(x\right)} \, dx$
$\Pi\!\left(n, m\right) = \int_{0}^{\pi / 2} \frac{1}{\left(1 - n \sin^{2}\!\left(x\right)\right) \sqrt{1 - m \sin^{2}\!\left(x\right)}} \, dx$

### Algebraic forms of the complete integrals

$K(m) = \int_{0}^{1} \frac{1}{\sqrt{1 - {x}^{2}} \sqrt{1 - m {x}^{2}}} \, dx$
$E(m) = \int_{0}^{1} \frac{\sqrt{1 - m {x}^{2}}}{\sqrt{1 - {x}^{2}}} \, dx$
$\Pi\!\left(n, m\right) = \int_{0}^{1} \frac{1}{\left(1 - n {x}^{2}\right) \sqrt{1 - {x}^{2}} \sqrt{1 - m {x}^{2}}} \, dx$
$K(m) = \int_{1}^{\infty} \frac{1}{\sqrt{{x}^{2} - 1} \sqrt{{x}^{2} - m}} \, dx$

### Trigonometric forms of the incomplete integrals

$F\!\left(\phi, m\right) = \int_{0}^{\phi} \frac{1}{\sqrt{1 - m \sin^{2}\!\left(x\right)}} \, dx$
$E\!\left(\phi, m\right) = \int_{0}^{\phi} \sqrt{1 - m \sin^{2}\!\left(x\right)} \, dx$
$\Pi\!\left(n, \phi, m\right) = \int_{0}^{\phi} \frac{1}{\left(1 - n \sin^{2}\!\left(x\right)\right) \sqrt{1 - m \sin^{2}\!\left(x\right)}} \, dx$

### Algebraic forms of the incomplete integrals

$F\!\left(\phi, m\right) = \int_{0}^{\sin(\phi)} \frac{1}{\sqrt{1 - {x}^{2}} \sqrt{1 - m {x}^{2}}} \, dx$
$E\!\left(\phi, m\right) = \int_{0}^{\sin(\phi)} \frac{\sqrt{1 - m {x}^{2}}}{\sqrt{1 - {x}^{2}}} \, dx$
$\Pi\!\left(n, \phi, m\right) = \int_{0}^{\sin(\phi)} \frac{1}{\left(1 - n {x}^{2}\right) \sqrt{1 - {x}^{2}} \sqrt{1 - m {x}^{2}}} \, dx$

## Specific values

### Complete elliptic integrals

$K(0) = \frac{\pi}{2}$
$E(0) = \frac{\pi}{2}$
$K(1) = \infty$
$E(1) = 1$
$K(-1) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}}$
$K\!\left(\frac{1}{2}\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{\pi}}$
$K(2) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}} \left(1 - i\right)$
$E(-1) = \sqrt{2} \left(\frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{8 \sqrt{\pi}} + \frac{{\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}\right)$
$E\!\left(\frac{1}{2}\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{8 \sqrt{\pi}} + \frac{{\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}$
$E(2) = \frac{\sqrt{2} {\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}} \left(1 + i\right)$
$K\!\left({\left(3 - 2 \sqrt{2}\right)}^{2}\right) = \frac{\left(2 + \sqrt{2}\right) {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{16 \sqrt{\pi}}$
$K\!\left(\frac{4 - 3 \sqrt{2}}{8}\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \cdot {2}^{1 / 4} \sqrt{\pi}}$
$K\!\left(\frac{1 + \sqrt{3} i}{2}\right) = \frac{{e}^{i \pi / 12} \cdot {3}^{1 / 4} {\left(\Gamma\!\left(\frac{1}{3}\right)\right)}^{3}}{{2}^{7 / 3} \pi}$
$K\!\left(\frac{1 - \sqrt{3} i}{2}\right) = \frac{{e}^{-i \pi / 12} \cdot {3}^{1 / 4} {\left(\Gamma\!\left(\frac{1}{3}\right)\right)}^{3}}{{2}^{7 / 3} \pi}$
$K\!\left(4 \sqrt{3} - 7\right) = \frac{\sqrt{3 + 2 \sqrt{3}} {\left(\Gamma\!\left(\frac{1}{3}\right)\right)}^{3}}{{2}^{10 / 3} \pi}$
$K\!\left(\frac{1}{2} - \frac{\sqrt{3}}{4}\right) = \frac{{3}^{1 / 4} {\left(\Gamma\!\left(\frac{1}{3}\right)\right)}^{3}}{4 \cdot {2}^{1 / 3} \pi}$
$\Pi\!\left(0, 0\right) = \frac{\pi}{2}$
$\Pi\!\left(0, 1\right) = \infty$
$\Pi\!\left(1, 0\right) = {\tilde \infty}$
$\Pi\!\left(0, \frac{1}{2}\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{\pi}}$
$\Pi\!\left(\frac{1}{2}, 0\right) = \frac{\pi \sqrt{2}}{2}$
$\Pi\!\left(\frac{1}{2}, \frac{1}{2}\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{\pi}} + \frac{2 {\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}$
$\Pi\!\left(1, m\right) = {\tilde \infty}$
$\Pi\!\left(n, 1\right) = \begin{cases} {\left(1 - n\right)}^{-1} \infty, & n \ne 1\\{\tilde \infty}, & n = 1\\ \end{cases}$
$\Pi\!\left(n, 0\right) = \frac{\pi}{2 \sqrt{1 - n}}$
$\Pi\!\left(0, m\right) = K(m)$
$\Pi\!\left(m, m\right) = \frac{E(m)}{1 - m}$

### Incomplete integral of the first kind

$F\!\left(0, 0\right) = 0$
$F\!\left(0, m\right) = 0$
$F\!\left(\phi, 0\right) = \phi$
$F\!\left(\frac{\pi}{2}, m\right) = K(m)$
$F\!\left(\frac{-\pi}{2}, m\right) = -K(m)$
$F\!\left(\frac{\pi k}{2}, m\right) = k K(m)$
$F\!\left(\frac{\pi}{2}, 0\right) = \frac{\pi}{2}$
$F\!\left(\frac{\pi}{2}, -1\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}}$
$F\!\left(\frac{\pi}{2}, 1\right) = \infty$
$F\!\left(\frac{-\pi}{2}, 1\right) = -\infty$
$F\!\left(\frac{\pi}{3}, 1\right) = \log\!\left(2 + \sqrt{3}\right)$
$F\!\left(\frac{\pi}{4}, 1\right) = \log\!\left(1 + \sqrt{2}\right)$
$F\!\left(\frac{\pi}{6}, 1\right) = \frac{\log(3)}{2}$
$F\!\left(\phi, 1\right) = \begin{cases} \log\!\left(\frac{1 + \sin(\phi)}{\cos(\phi)}\right), & \frac{-\pi}{2} \le \operatorname{Re}(\phi) \le \frac{\pi}{2} \;\mathbin{\operatorname{and}}\; \phi \notin \left\{\frac{-\pi}{2}, \frac{\pi}{2}\right\}\\\operatorname{sgn}(\phi) \infty, & \phi \in \left\{\frac{-\pi}{2}, \frac{\pi}{2}\right\}\\{\tilde \infty}, & \text{otherwise}\\ \end{cases}$
$F\!\left(\frac{\pi}{4}, 2\right) = \frac{\sqrt{2} {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{8 \sqrt{\pi}}$
$F\!\left(\frac{\pi}{6}, 4\right) = \frac{K\!\left(\frac{1}{4}\right)}{2}$
$F\!\left(\operatorname{asin}\!\left(\frac{1}{\sqrt{m}}\right), m\right) = \frac{K\!\left(\frac{1}{m}\right)}{\sqrt{m}}$

### Incomplete integral of the second kind

$E\!\left(0, 0\right) = 0$
$E\!\left(0, m\right) = 0$
$E\!\left(\phi, 0\right) = \phi$
$E\!\left(\frac{\pi}{2}, m\right) = E(m)$
$E\!\left(\frac{-\pi}{2}, m\right) = -E(m)$
$E\!\left(\frac{\pi k}{2}, m\right) = k E(m)$
$E\!\left(\phi, 1\right) = \sin(\phi)$
$E\!\left(\phi, 1\right) = {\left(-1\right)}^{\left\lfloor \operatorname{Re}(\phi) / \pi + 1 / 2 \right\rfloor} \sin(\phi) + 2 \left\lfloor \frac{\operatorname{Re}(\phi)}{\pi} + \frac{1}{2} \right\rfloor$
$E\!\left(\frac{\pi}{2}, 0\right) = \frac{\pi}{2}$
$E\!\left(\frac{\pi}{2}, -1\right) = \sqrt{2} \left(\frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{8 \sqrt{\pi}} + \frac{{\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}\right)$
$E\!\left(\frac{\pi}{2}, 1\right) = 1$
$E\!\left(\frac{-\pi}{2}, 1\right) = -1$
$E\!\left(\frac{\pi k}{2}, 1\right) = k$
$E\!\left(\frac{\pi}{3}, 1\right) = \frac{\sqrt{3}}{2}$
$E\!\left(\frac{\pi}{6}, 1\right) = \frac{1}{2}$
$E\!\left(\frac{\pi}{4}, 2\right) = \frac{\sqrt{2} {\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}$
$E\!\left(\frac{\pi}{6}, 4\right) = 2 E\!\left(\frac{1}{4}\right) - \frac{3}{2} K\!\left(\frac{1}{4}\right)$
$E\!\left(\operatorname{asin}\!\left(\frac{1}{\sqrt{m}}\right), m\right) = \sqrt{m} \left(E\!\left(\frac{1}{m}\right) - \left(1 - \frac{1}{m}\right) K\!\left(\frac{1}{m}\right)\right)$

## Functional equations

### Conjugate symmetry

$K\!\left(\overline{m}\right) = \overline{K(m)}$
$E\!\left(\overline{m}\right) = \overline{E(m)}$

### Odd symmetry

$F\!\left(-\phi, m\right) = -F\!\left(\phi, m\right)$
$E\!\left(-\phi, m\right) = -E\!\left(\phi, m\right)$
$\Pi\!\left(n, -\phi, m\right) = -\Pi\!\left(n, \phi, m\right)$

### Quasi-periodicity

$F\!\left(\phi + k \pi, m\right) = F\!\left(\phi, m\right) + 2 k K(m)$
$E\!\left(\phi + k \pi, m\right) = E\!\left(\phi, m\right) + 2 k E(m)$
$\Pi\!\left(n, \phi + k \pi, m\right) = \Pi\!\left(n, \phi, m\right) + 2 k \Pi\!\left(n, m\right)$

## Representation by other functions

### Hypergeometric functions

$K(m) = \frac{\pi}{2} \,{}_2F_1\!\left(\frac{1}{2}, \frac{1}{2}, 1, m\right)$
$E(m) = \frac{\pi}{2} \,{}_2F_1\!\left(-\frac{1}{2}, \frac{1}{2}, 1, m\right)$
$2 E(m) - K(m) = \frac{\pi}{2} \,{}_2F_1\!\left(-\frac{1}{2}, \frac{3}{2}, 1, m\right)$

### Arithmetic-geometric mean

$K(m) = \frac{\pi}{2 \operatorname{agm}\!\left(1, \sqrt{1 - m}\right)}$

### Carlson symmetric elliptic integrals

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