# Fungrim entry: 5f84d9

$\Pi\!\left(n, \phi + k \pi, m\right) = \Pi\!\left(n, \phi, m\right) + 2 k \Pi\!\left(n, m\right)$
Assumptions:$n \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \phi \in \mathbb{C} \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \ne 1 \;\mathbin{\operatorname{and}}\; m \ne 1$
TeX:
\Pi\!\left(n, \phi + k \pi, m\right) = \Pi\!\left(n, \phi, m\right) + 2 k \Pi\!\left(n, m\right)

n \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \phi \in \mathbb{C} \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \ne 1 \;\mathbin{\operatorname{and}}\; m \ne 1
Definitions:
Fungrim symbol Notation Short description
IncompleteEllipticPi$\Pi\!\left(n, \phi, m\right)$ Legendre incomplete elliptic integral of the third kind
Pi$\pi$ The constant pi (3.14...)
EllipticPi$\Pi\!\left(n, m\right)$ Legendre complete elliptic integral of the third kind
CC$\mathbb{C}$ Complex numbers
ZZ$\mathbb{Z}$ Integers
Source code for this entry:
Entry(ID("5f84d9"),
Formula(Equal(IncompleteEllipticPi(n, Add(phi, Mul(k, Pi)), m), Add(IncompleteEllipticPi(n, phi, m), Mul(Mul(2, k), EllipticPi(n, m))))),
Variables(n, phi, m, k),
Assumptions(And(Element(n, CC), Element(phi, CC), Element(m, CC), Element(k, ZZ), NotEqual(n, 1), NotEqual(m, 1))))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC