DLMF:10.23.E19 (Q3473)

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DLMF:10.23.E19
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    Statements

    Defining formula
    a m = 2 ( J ν + 1 ⁡ ( j ν , m ) ) 2 ⁢ ∫ 0 1 t ⁢ f ⁢ ( t ) ⁢ J ν ⁡ ( j ν , m ⁢ t ) ⁢ d t , subscript 𝑎 𝑚 2 superscript Bessel-J 𝜈 1 zero-Bessel-J 𝜈 𝑚 2 superscript subscript 0 1 𝑡 𝑓 𝑡 Bessel-J 𝜈 subscript 𝑗 𝜈 𝑚 𝑡 𝑡 {\displaystyle{\displaystyle a_{m}=\frac{2}{(J_{\nu+1}\left(j_{\nu,m}\right))^% {2}}\int_{0}^{1}tf(t)J_{\nu}\left(j_{\nu,m}t\right)\mathrm{d}t,}}
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    DLMF id
    DLMF:10.23.E19
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    constraint
    ν ≥ - 1 2 𝜈 1 2 {\displaystyle{\displaystyle\nu\geq-\tfrac{1}{2}}}
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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.m2amdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abdec
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    Q11456
    Defining formula
    j ν , m zero-Bessel-J 𝜈 𝑚 {\displaystyle{\displaystyle j_{\NVar{\nu},\NVar{m}}}}
    xml-id
    C10.S21.SS1.p2.m11adec
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    Q11438
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C10.S1.XMD1.m1dec
    0 references
    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1gdec
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    DLMF:10.23.E11
    Defining formula
    a k subscript 𝑎 𝑘 {\displaystyle{\displaystyle a_{k}}}
    xml-id
    C10.S23.XMD2.m1cdec
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    Q11476
    Defining formula
    f ⁢ ( t ) 𝑓 𝑡 {\displaystyle{\displaystyle f(t)}}
    xml-id
    C10.S23.XMD3.m1ddec
    0 references
     
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