DLMF:11.5.E2 (Q3969)

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DLMF:11.5.E2
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    Statements

    Defining formula
    𝐊 ν ⁡ ( z ) = 2 ⁢ ( 1 2 ⁢ z ) ν π ⁢ Γ ⁡ ( ν + 1 2 ) ⁢ ∫ 0 ∞ e - z ⁢ t ⁢ ( 1 + t 2 ) ν - 1 2 ⁢ d t , associated-Struve-K 𝜈 𝑧 2 superscript 1 2 𝑧 𝜈 𝜋 Euler-Gamma 𝜈 1 2 superscript subscript 0 superscript 𝑒 𝑧 𝑡 superscript 1 superscript 𝑡 2 𝜈 1 2 𝑡 {\displaystyle{\displaystyle\mathbf{K}_{\nu}\left(z\right)=\frac{2(\tfrac{1}{2% }z)^{\nu}}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{1}{2}\right)}\int_{0}^{\infty}e^{-% zt}(1+t^{2})^{\nu-\frac{1}{2}}\mathrm{d}t,}}
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    DLMF id
    DLMF:11.5.E2
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    constraint
    ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}}
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    Symbols used
    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
    𝐊 ν ⁡ ( z ) associated-Struve-K 𝜈 𝑧 {\displaystyle{\displaystyle\mathbf{K}_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C11.S2.E5.m2adec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
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