DLMF:19.6.E3 (Q6146)

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DLMF:19.6.E3
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    Statements

    Defining formula
    Π ⁡ ( α 2 , 0 ) = π / ( 2 ⁢ 1 - α 2 ) , Π ⁡ ( 0 , k ) = K ⁡ ( k ) , formulae-sequence complete-elliptic-integral-third-kind-Pi superscript 𝛼 2 0 𝜋 2 1 superscript 𝛼 2 complete-elliptic-integral-third-kind-Pi 0 𝑘 complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle\Pi\left(\alpha^{2},0\right)=\pi/(2\sqrt{1-\alpha^% {2}}),\quad\Pi\left(0,k\right)=K\left(k\right),}}
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    DLMF id
    DLMF:19.6.E3
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    constraint
    - ∞ < α 2 < 1 superscript 𝛼 2 1 {\displaystyle{\displaystyle-\infty<\alpha^{2}<1}}
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    Symbols used
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
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    Q11600
    Defining formula
    K ⁡ ( k ) complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle K\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m1abdec
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    Q11816
    Defining formula
    Π ⁡ ( α 2 , k ) complete-elliptic-integral-third-kind-Pi superscript 𝛼 2 𝑘 {\displaystyle{\displaystyle\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m4aadec
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