DLMF:11.5.E8 (Q3975)

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DLMF:11.5.E8
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    Statements

    Defining formula
    ( 1 2 ⁢ x ) - ν - 1 ⁢ 𝐇 ν ⁡ ( x ) = - 1 2 ⁢ π ⁢ i ⁢ ∫ - i ⁢ ∞ i ⁢ ∞ π ⁢ csc ⁡ ( π ⁢ s ) Γ ⁡ ( 3 2 + s ) ⁢ Γ ⁡ ( 3 2 + ν + s ) ⁢ ( 1 4 ⁢ x 2 ) s ⁢ d s , superscript 1 2 𝑥 𝜈 1 Struve-H 𝜈 𝑥 1 2 𝜋 𝑖 superscript subscript 𝑖 𝑖 𝜋 𝜋 𝑠 Euler-Gamma 3 2 𝑠 Euler-Gamma 3 2 𝜈 𝑠 superscript 1 4 superscript 𝑥 2 𝑠 𝑠 {\displaystyle{\displaystyle(\tfrac{1}{2}x)^{-\nu-1}\mathbf{H}_{\nu}\left(x% \right)=-\frac{1}{2\pi i}\int_{-i\infty}^{i\infty}\frac{\pi\csc\left(\pi s% \right)}{\Gamma\left(\tfrac{3}{2}+s\right)\Gamma\left(\tfrac{3}{2}+\nu+s\right% )}(\tfrac{1}{4}x^{2})^{s}\mathrm{d}s,}}
    0 references
    DLMF id
    DLMF:11.5.E8
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    constraint
    ℜ ⁡ ν > - 1 𝜈 1 {\displaystyle{\displaystyle\Re\nu>-1}}
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    x > 0 𝑥 0 {\displaystyle{\displaystyle x>0}}
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    ℜ ⁡ ν > - 1 𝜈 1 {\displaystyle{\displaystyle\Re\nu>-1}}
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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.m2aedec
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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.m2aadec
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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.m2agdec
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    cosecant function
    Defining formula
    csc ⁡ z 𝑧 {\displaystyle{\displaystyle\csc\NVar{z}}}
    xml-id
    C4.S14.E5.m2adec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1agdec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3agdec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1afdec
    0 references
     
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