DLMF:22.16.E31 (Q7151)

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DLMF:22.16.E31
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    Statements

    Defining formula
    E ⁡ ( am ⁡ ( x , k ) , k ) = ℰ ⁡ ( x , k ) , elliptic-integral-second-kind-E Jacobi-elliptic-amplitude 𝑥 𝑘 𝑘 Jacobi-Epsilon 𝑥 𝑘 {\displaystyle{\displaystyle E\left(\operatorname{am}\left(x,k\right),k\right)% =\mathcal{E}\left(x,k\right),}}
    0 references
    DLMF id
    DLMF:22.16.E31
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    constraint
    - K ≤ x ≤ K 𝐾 𝑥 𝐾 {\displaystyle{\displaystyle-K\leq x\leq K}}
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    Symbols used
    Jacobi’s epsilon function
    Defining formula
    ℰ ⁡ ( x , k ) Jacobi-Epsilon 𝑥 𝑘 {\displaystyle{\displaystyle\mathcal{E}\left(\NVar{x},\NVar{k}\right)}}
    xml-id
    C22.S16.E14.m2aqdec
    0 references
    Jacobi’s amplitude function
    Defining formula
    am ⁡ ( x , k ) Jacobi-elliptic-amplitude 𝑥 𝑘 {\displaystyle{\displaystyle\operatorname{am}\left(\NVar{x},\NVar{k}\right)}}
    xml-id
    C22.S16.E1.m2aldec
    0 references
    Q11600
    Defining formula
    K ⁡ ( k ) complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle K\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m1aedec
    0 references
    Legendre’s incomplete elliptic integral of the second kind
    Defining formula
    E ⁡ ( ϕ , k ) elliptic-integral-second-kind-E italic-ϕ 𝑘 {\displaystyle{\displaystyle E\left(\NVar{\phi},\NVar{k}\right)}}
    xml-id
    C19.S2.E5.m2adec
    0 references
    Q12003
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C22.S1.XMD1.m1addec
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    Q11984
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C22.S1.XMD4.m1abdec
    0 references
     
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