DLMF:19.9.E15 (Q6258)

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DLMF:19.9.E15
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    Statements

    Defining formula
    1 < F ⁡ ( ϕ , k ) / ( ( sin ⁡ ϕ ) ⁢ ln ⁡ ( 4 Δ + cos ⁡ ϕ ) ) < 4 2 + ( 1 + k 2 ) ⁢ sin 2 ⁡ ϕ . 1 elliptic-integral-first-kind-F italic-ϕ 𝑘 italic-ϕ 4 Δ italic-ϕ 4 2 1 superscript 𝑘 2 2 italic-ϕ {\displaystyle{\displaystyle 1<F\left(\phi,k\right)\bigg{/}\left((\sin\phi)\ln% \left(\frac{4}{\Delta+\cos\phi}\right)\right)<\frac{4}{2+(1+k^{2}){\sin^{2}}% \phi}.}}
    0 references
    DLMF id
    DLMF:19.9.E15
    0 references
    Symbols used
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aadec
    0 references
    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.m2abdec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2addec
    0 references
     
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