DLMF:11.5.E1 (Q3968)

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DLMF:11.5.E1
No description defined

    Statements

    Defining formula
    𝐇 ν ⁡ ( z ) = 2 ⁢ ( 1 2 ⁢ z ) ν π ⁢ Γ ⁡ ( ν + 1 2 ) ⁢ ∫ 0 1 ( 1 - t 2 ) ν - 1 2 ⁢ sin ⁡ ( z ⁢ t ) ⁢ d t = 2 ⁢ ( 1 2 ⁢ z ) ν π ⁢ Γ ⁡ ( ν + 1 2 ) ⁢ ∫ 0 π / 2 sin ⁡ ( z ⁢ cos ⁡ θ ) ⁢ ( sin ⁡ θ ) 2 ⁢ ν ⁢ d θ , Struve-H 𝜈 𝑧 2 superscript 1 2 𝑧 𝜈 𝜋 Euler-Gamma 𝜈 1 2 superscript subscript 0 1 superscript 1 superscript 𝑡 2 𝜈 1 2 𝑧 𝑡 𝑡 2 superscript 1 2 𝑧 𝜈 𝜋 Euler-Gamma 𝜈 1 2 superscript subscript 0 𝜋 2 𝑧 𝜃 superscript 𝜃 2 𝜈 𝜃 {\displaystyle{\displaystyle\mathbf{H}_{\nu}\left(z\right)=\frac{2(\tfrac{1}{2% }z)^{\nu}}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{1}{2}\right)}\int_{0}^{1}(1-t^{2})% ^{\nu-\frac{1}{2}}\sin\left(zt\right)\mathrm{d}t=\frac{2(\tfrac{1}{2}z)^{\nu}}% {\sqrt{\pi}\Gamma\left(\nu+\tfrac{1}{2}\right)}\int_{0}^{\pi/2}\sin\left(z\cos% \theta\right)(\sin\theta)^{2\nu}\mathrm{d}\theta,}}
    0 references
    DLMF id
    DLMF:11.5.E1
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    constraint
    ℜ ⁡ ν > - 1 2 𝜈 1 2 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2}}}
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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.m2adec
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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.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1adec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2adec
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    Q11491
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C11.S1.XMD2.m1dec
    0 references
    Q11492
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C11.S1.XMD3.m1dec
    0 references
     
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