Fungrim entry: 685126

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

Assumptions:

\phi \in \mathbb{C} \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; m \ne 1

TeX:

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

\phi \in \mathbb{C} \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; m \ne 1

Definitions:

Fungrim symbol	Notation	Short description
`IncompleteEllipticF`	$F\!\left(\phi, m\right)$	Legendre incomplete elliptic integral of the first kind
`Pi`	$\pi$	The constant pi (3.14...)
`EllipticK`	$K(m)$	Legendre complete elliptic integral of the first kind
`CC`	$\mathbb{C}$	Complex numbers
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("685126"),
    Formula(Equal(IncompleteEllipticF(Add(phi, Mul(k, Pi)), m), Add(IncompleteEllipticF(phi, m), Mul(Mul(2, k), EllipticK(m))))),
    Variables(phi, m, k),
    Assumptions(And(Element(phi, CC), Element(m, CC), Element(k, ZZ), NotEqual(m, 1))))

Topics using this entry

Legendre elliptic integrals