DLMF:19.7#Ex11 (Q6192)

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DLMF:19.7#Ex11
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    Statements

    Defining formula
    Π ⁡ ( ϕ , α 2 , k 1 ) = k ⁢ Π ⁡ ( β , k 2 ⁢ α 2 , k ) , elliptic-integral-third-kind-Pi italic-ϕ superscript 𝛼 2 subscript 𝑘 1 𝑘 elliptic-integral-third-kind-Pi 𝛽 superscript 𝑘 2 superscript 𝛼 2 𝑘 {\displaystyle{\displaystyle\Pi\left(\phi,\alpha^{2},k_{1}\right)=k\Pi\left(% \beta,k^{2}\alpha^{2},k\right),}}
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    DLMF id
    DLMF:19.7#Ex11
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    constraint
    sin ⁡ β = k 1 ⁢ sin ⁡ ϕ ≤ 1 𝛽 subscript 𝑘 1 italic-ϕ 1 {\displaystyle{\displaystyle\sin\beta=k_{1}\sin\phi\leq 1}}
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    k 1 = 1 / k subscript 𝑘 1 1 𝑘 {\displaystyle{\displaystyle k_{1}=1/k}}
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    sin ⁡ β = k 1 ⁢ sin ⁡ ϕ ≤ 1 𝛽 subscript 𝑘 1 italic-ϕ 1 {\displaystyle{\displaystyle\sin\beta=k_{1}\sin\phi\leq 1}}
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