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

Legendre elliptic integrals