DLMF:10.23.E9 (Q3463)

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DLMF:10.23.E9
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    Statements

    Defining formula
    e i ⁢ v ⁢ cos ⁡ α = Γ ⁡ ( ν ) ( 1 2 ⁢ v ) ν ⁢ ∑ k = 0 ∞ ( ν + k ) ⁢ i k ⁢ J ν + k ⁡ ( v ) ⁢ C k ( ν ) ⁡ ( cos ⁡ α ) , superscript 𝑒 𝑖 𝑣 𝛼 Euler-Gamma 𝜈 superscript 1 2 𝑣 𝜈 superscript subscript 𝑘 0 𝜈 𝑘 superscript 𝑖 𝑘 Bessel-J 𝜈 𝑘 𝑣 ultraspherical-Gegenbauer-polynomial 𝜈 𝑘 𝛼 {\displaystyle{\displaystyle e^{iv\cos\alpha}=\frac{\Gamma\left(\nu\right)}{(% \tfrac{1}{2}v)^{\nu}}\*\sum_{k=0}^{\infty}(\nu+k)i^{k}J_{\nu+k}\left(v\right)C% ^{(\nu)}_{k}\left(\cos\alpha\right),}}
    0 references
    DLMF id
    DLMF:10.23.E9
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    constraint
    ν ≠ 0 , - 1 , … 𝜈 0 1 … {\displaystyle{\displaystyle\nu\neq 0,-1,\dots}}
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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.m2afdec
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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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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2acdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2abdec
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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.m2aadec
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    Q11428
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C10.S1.XMD3.m1gdec
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    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1ddec
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