DLMF:14.5.E21 (Q4733)

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DLMF:14.5.E21
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    Statements

    Defining formula
    𝖯 - 1 2 ⁡ ( cos ⁡ θ ) = 2 π ⁢ K ⁡ ( sin ⁡ ( 1 2 ⁢ θ ) ) , shorthand-Ferrers-Legendre-P-first-kind 1 2 𝜃 2 𝜋 complete-elliptic-integral-first-kind-K 1 2 𝜃 {\displaystyle{\displaystyle\mathsf{P}_{-\frac{1}{2}}\left(\cos\theta\right)=% \frac{2}{\pi}K\left(\sin\left(\tfrac{1}{2}\theta\right)\right),}}
    0 references
    DLMF id
    DLMF:14.5.E21
    0 references
    Symbols used
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aldec
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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.m1aadec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2agdec
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    Q11595
    Defining formula
    𝖯 ν ⁡ ( x ) = 𝖯 ν 0 ⁡ ( x ) shorthand-Ferrers-Legendre-P-first-kind 𝜈 𝑥 Ferrers-Legendre-P-first-kind 0 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=% \mathsf{P}^{0}_{\nu}\left(x\right)}}
    xml-id
    C14.S2.SS2.p2.m2acdec
    0 references
     
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