DLMF:19.9.E11 (Q6254)

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DLMF:19.9.E11
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    Statements

    Defining formula
    ϕ ≤ F ⁡ ( ϕ , k ) ≤ min ⁡ ( ϕ / Δ , gd - 1 ⁡ ( ϕ ) ) , italic-ϕ elliptic-integral-first-kind-F italic-ϕ 𝑘 italic-ϕ Δ inverse-Gudermannian italic-ϕ {\displaystyle{\displaystyle\phi\leq F\left(\phi,k\right)\leq\min(\phi/\Delta,% {\operatorname{gd}^{-1}}\left(\phi\right)),}}
    0 references
    DLMF id
    DLMF:19.9.E11
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    Symbols used
    Legendre’s incomplete elliptic integral of the first kind
    Defining formula
    F ⁡ ( ϕ , k ) elliptic-integral-first-kind-F italic-ϕ 𝑘 {\displaystyle{\displaystyle F\left(\NVar{\phi},\NVar{k}\right)}}
    xml-id
    C19.S2.E4.m2adec
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    inverse Gudermannian function
    Defining formula
    gd - 1 ⁡ ( x ) inverse-Gudermannian 𝑥 {\displaystyle{\displaystyle{\operatorname{gd}^{-1}}\left(\NVar{x}\right)}}
    xml-id
    C4.S23.E41.m2adec
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    Q11807
    Defining formula
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    xml-id
    C19.S1.XMD4.m1dec
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    Q11808
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1idec
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    Q11839
    Defining formula
    Δ Δ {\displaystyle{\displaystyle\Delta}}
    xml-id
    C19.S9.XMD2.m1dec
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