DLMF:11.4.E20 (Q3952)

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DLMF:11.4.E20
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    Statements

    Defining formula
    𝐇 ν ⁡ ( z ) = ( 1 2 ⁢ z ) ν + 1 2 Γ ⁡ ( ν + 1 2 ) ⁢ ∑ k = 0 ∞ ( 1 2 ⁢ z ) k k ! ⁢ ( k + ν + 1 2 ) ⁢ J k + 1 2 ⁡ ( z ) , Struve-H 𝜈 𝑧 superscript 1 2 𝑧 𝜈 1 2 Euler-Gamma 𝜈 1 2 superscript subscript 𝑘 0 superscript 1 2 𝑧 𝑘 𝑘 𝑘 𝜈 1 2 Bessel-J 𝑘 1 2 𝑧 {\displaystyle{\displaystyle\mathbf{H}_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}% z)^{\nu+\frac{1}{2}}}{\Gamma\left(\nu+\tfrac{1}{2}\right)}\sum_{k=0}^{\infty}% \frac{(\tfrac{1}{2}z)^{k}}{k!(k+\nu+\tfrac{1}{2})}J_{k+\frac{1}{2}}\left(z% \right),}}
    0 references
    DLMF id
    DLMF:11.4.E20
    0 references
    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.m2acdec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2abdec
    0 references
    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.m2ajdec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5addec
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    Q11491
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C11.S1.XMD2.m1rdec
    0 references
    Q11492
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C11.S1.XMD3.m1gdec
    0 references
    Q11497
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C11.S1.XMD5.m1bdec
    0 references
     
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