DLMF:11.4.E18 (Q3950)

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DLMF:11.4.E18
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    Statements

    Defining formula
    𝐇 ν ⁡ ( z ) = 4 π 1 / 2 ⁢ Γ ⁡ ( ν + 1 2 ) ⁢ ∑ k = 0 ∞ ( 2 ⁢ k + ν + 1 ) ⁢ Γ ⁡ ( k + ν + 1 ) k ! ⁢ ( 2 ⁢ k + 1 ) ⁢ ( 2 ⁢ k + 2 ⁢ ν + 1 ) ⁢ J 2 ⁢ k + ν + 1 ⁡ ( z ) , Struve-H 𝜈 𝑧 4 superscript 𝜋 1 2 Euler-Gamma 𝜈 1 2 superscript subscript 𝑘 0 2 𝑘 𝜈 1 Euler-Gamma 𝑘 𝜈 1 𝑘 2 𝑘 1 2 𝑘 2 𝜈 1 Bessel-J 2 𝑘 𝜈 1 𝑧 {\displaystyle{\displaystyle\mathbf{H}_{\nu}\left(z\right)=\frac{4}{\pi^{1/2}% \Gamma\left(\nu+\tfrac{1}{2}\right)}\*\sum_{k=0}^{\infty}\frac{(2k+\nu+1)% \Gamma\left(k+\nu+1\right)}{k!(2k+1)(2k+2\nu+1)}J_{2k+\nu+1}\left(z\right),}}
    0 references
    DLMF id
    DLMF:11.4.E18
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    constraint
    ν ≠ - 1 , - 2 , - 3 , … 𝜈 1 2 3 … {\displaystyle{\displaystyle\nu\neq-1,-2,-3,\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.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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    Struve function
    Defining formula
    𝐇 ν ⁡ ( z ) Struve-H 𝜈 𝑧 {\displaystyle{\displaystyle\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C11.S2.E1.m2ahdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2amdec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5abdec
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    Q11491
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C11.S1.XMD2.m1pdec
    0 references
    Q11492
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C11.S1.XMD3.m1edec
    0 references
    Q11497
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C11.S1.XMD5.m1dec
    0 references
     
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