DLMF:10.23.E7 (Q3461)

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DLMF:10.23.E7
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    Statements

    test
    𝒞 ν ⁡ ( w ) ⁢ cos sin ⁢ ( ν ⁢ χ ) = ∑ k = - ∞ ∞ 𝒞 ν + k ⁡ ( u ) ⁢ J k ⁡ ( v ) ⁢ cos sin ⁢ ( k ⁢ α ) , cylinder-function 𝜈 𝑤 selection 𝜈 𝜒 superscript subscript 𝑘 cylinder-function 𝜈 𝑘 𝑢 Bessel-J 𝑘 𝑣 selection 𝑘 𝛼 {\displaystyle{\displaystyle\mathscr{C}_{\nu}\left(w\right)\selection{\cos\\ \sin}(\nu\chi)=\sum_{k=-\infty}^{\infty}\mathscr{C}_{\nu+k}\left(u\right)J_{k}% \left(v\right)\selection{\cos\\ \sin}(k\alpha),}}
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    DLMF id
    DLMF:10.23.E7
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    constraint
    | v ⁢ e ± i ⁢ α | < | u | 𝑣 superscript 𝑒 plus-or-minus 𝑖 𝛼 𝑢 {\displaystyle{\displaystyle|ve^{\pm i\alpha}|<|u|}}
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    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.m2addec
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    cosine function
    test
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aadec
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    Q11430
    test
    𝒞 ν ⁡ ( z ) cylinder-function 𝜈 𝑧 {\displaystyle{\displaystyle\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.Px5.p1.m3abdec
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    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
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    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
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    sine function
    test
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aadec
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