DLMF:18.12.E6 (Q5648): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: gamma function / rank
 
Normal rank
Property / Symbols used: gamma function / qualifier
 
test:

Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}

\EulerGamma@{\NVar{z}}
Property / Symbols used: gamma function / qualifier
 
xml-id: C5.S2.E1.m2aadec

Revision as of 14:31, 2 January 2020

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DLMF:18.12.E6
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    Statements

    test
    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},}}
    0 references
    DLMF id
    DLMF:18.12.E6
    0 references
    constraint
    0 ≤ θ ≤ π 0 𝜃 𝜋 {\displaystyle{\displaystyle 0\leq\theta\leq\pi}}
    0 references
    Symbols used
    Bessel function of the first kind
    test
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2aadec
    0 references
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aadec
    0 references
     
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