DLMF:14.20.E10 (Q4931)

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DLMF:14.20.E10
No description defined

    Statements

    Defining formula
    𝖯 - 1 2 + i ⁢ τ ⁡ ( cos ⁡ θ ) = 1 + 4 ⁢ τ 2 + 1 2 2 2 ⁢ sin 2 ⁡ ( 1 2 ⁢ θ ) + ( 4 ⁢ τ 2 + 1 2 ) ⁢ ( 4 ⁢ τ 2 + 3 2 ) 2 2 ⋅ 4 2 ⁢ sin 4 ⁡ ( 1 2 ⁢ θ ) + ⋯ , shorthand-Ferrers-Legendre-P-first-kind 1 2 𝑖 𝜏 𝜃 1 4 superscript 𝜏 2 superscript 1 2 superscript 2 2 2 1 2 𝜃 4 superscript 𝜏 2 superscript 1 2 4 superscript 𝜏 2 superscript 3 2 ⋅ superscript 2 2 superscript 4 2 4 1 2 𝜃 ⋯ {\displaystyle{\displaystyle\mathsf{P}_{-\frac{1}{2}+i\tau}\left(\cos\theta% \right)=1+\frac{4\tau^{2}+1^{2}}{2^{2}}{\sin^{2}}\left(\tfrac{1}{2}\theta% \right)+\frac{\left(4\tau^{2}+1^{2}\right)\left(4\tau^{2}+3^{2}\right)}{2^{2}% \cdot 4^{2}}{\sin^{4}}\left(\tfrac{1}{2}\theta\right)+\cdots,}}
    0 references
    DLMF id
    DLMF:14.20.E10
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    constraint
    0 ≤ θ ≤ π 0 𝜃 𝜋 {\displaystyle{\displaystyle 0\leq\theta\leq\pi}}
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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.m2afdec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2addec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2apdec
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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.m2aadec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2addec
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    Q11630
    Defining formula
    τ 𝜏 {\displaystyle{\displaystyle\tau}}
    xml-id
    C14.S1.XMD3.m1qdec
    0 references
     
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