DLMF:19.2.E6 (Q6087)

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DLMF:19.2.E6
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    Statements

    Defining formula
    D ⁡ ( ϕ , k ) = ∫ 0 ϕ sin 2 ⁡ θ ⁢ d θ 1 - k 2 ⁢ sin 2 ⁡ θ = ∫ 0 sin ⁡ ϕ t 2 ⁢ d t 1 - t 2 ⁢ 1 - k 2 ⁢ t 2 = ( F ⁡ ( ϕ , k ) - E ⁡ ( ϕ , k ) ) / k 2 . elliptic-integral-third-kind-D italic-ϕ 𝑘 superscript subscript 0 italic-ϕ 2 𝜃 𝜃 1 superscript 𝑘 2 2 𝜃 superscript subscript 0 italic-ϕ superscript 𝑡 2 𝑡 1 superscript 𝑡 2 1 superscript 𝑘 2 superscript 𝑡 2 elliptic-integral-first-kind-F italic-ϕ 𝑘 elliptic-integral-second-kind-E italic-ϕ 𝑘 superscript 𝑘 2 {\displaystyle{\displaystyle D\left(\phi,k\right)=\int_{0}^{\phi}\frac{{\sin^{% 2}}\theta\mathrm{d}\theta}{\sqrt{1-k^{2}{\sin^{2}}\theta}}=\int_{0}^{\sin\phi}% \frac{t^{2}\mathrm{d}t}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}}=(F\left(\phi,k% \right)-E\left(\phi,k\right))/k^{2}.}}
    0 references
    DLMF id
    DLMF:19.2.E6
    0 references
    Symbols used
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1addec
    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.m2aadec
    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.m2aadec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3addec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2abdec
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    Q11807
    Defining formula
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    xml-id
    C19.S1.XMD4.m1bdec
    0 references
    Q11808
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1bdec
    0 references
     
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