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Fungrim entry: 5f84d9

Π ⁣(n,ϕ+kπ,m)=Π ⁣(n,ϕ,m)+2kΠ ⁣(n,m)\Pi\!\left(n, \phi + k \pi, m\right) = \Pi\!\left(n, \phi, m\right) + 2 k \Pi\!\left(n, m\right)
Assumptions:nC  and  ϕC  and  mC  and  kZ  and  n1  and  m1n \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
\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
Fungrim symbol Notation Short description
IncompleteEllipticPiΠ ⁣(n,ϕ,m)\Pi\!\left(n, \phi, m\right) Legendre incomplete elliptic integral of the third kind
Piπ\pi The constant pi (3.14...)
EllipticPiΠ ⁣(n,m)\Pi\!\left(n, m\right) Legendre complete elliptic integral of the third kind
CCC\mathbb{C} Complex numbers
ZZZ\mathbb{Z} Integers
Source code for this entry:
    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))))

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2021-03-15 19:12:00.328586 UTC