DLMF:18.12.E6 (Q5648)

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DLMF:18.12.E6
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    Statements

    Defining formula
    Γ ⁡ ( λ + 1 2 ) ⁢ e z ⁢ cos ⁡ θ ⁢ ( 1 2 ⁢ z ⁢ sin ⁡ θ ) 1 2 - λ ⁢ J λ - 1 2 ⁡ ( z ⁢ sin ⁡ θ ) = ∑ n = 0 ∞ C n ( λ ) ⁡ ( cos ⁡ θ ) ( 2 ⁢ λ ) n ⁢ z n , Euler-Gamma 𝜆 1 2 superscript 𝑒 𝑧 𝜃 superscript 1 2 𝑧 𝜃 1 2 𝜆 Bessel-J 𝜆 1 2 𝑧 𝜃 superscript subscript 𝑛 0 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝜃 Pochhammer 2 𝜆 𝑛 superscript 𝑧 𝑛 {\displaystyle{\displaystyle\Gamma\left(\lambda+\tfrac{1}{2}\right)e^{z\cos% \theta}(\tfrac{1}{2}z\sin\theta)^{\frac{1}{2}-\lambda}J_{\lambda-\frac{1}{2}}% \left(z\sin\theta\right)=\sum_{n=0}^{\infty}\frac{C^{(\lambda)}_{n}\left(\cos% \theta\right)}{{\left(2\lambda\right)_{n}}}z^{n},}}
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    DLMF id
    DLMF:18.12.E6
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    constraint
    0 ≤ θ ≤ π 0 𝜃 𝜋 {\displaystyle{\displaystyle 0\leq\theta\leq\pi}}
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    Symbols used
    Bessel function of the first kind
    Defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2aadec
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    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aadec
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    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1abdec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
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    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2adec
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    Q11475
    Defining formula
    C n ( λ ) ⁡ ( x ) ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 {\displaystyle{\displaystyle C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}% \right)}}
    xml-id
    C18.S3.T1.t1.r3.m2abdec
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    Q11750
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C18.S1.XMD2.m1edec
    0 references
     
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