DLMF:11.2.E2 (Q3917)

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DLMF:11.2.E2
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    Statements

    test
    𝐋 ν ⁡ ( z ) = - i ⁢ e - 1 2 ⁢ π ⁢ i ⁢ ν ⁢ 𝐇 ν ⁡ ( i ⁢ z ) = ( 1 2 ⁢ z ) ν + 1 ⁢ ∑ n = 0 ∞ ( 1 2 ⁢ z ) 2 ⁢ n Γ ⁡ ( n + 3 2 ) ⁢ Γ ⁡ ( n + ν + 3 2 ) . modified-Struve-L 𝜈 𝑧 𝑖 superscript 𝑒 1 2 𝜋 𝑖 𝜈 Struve-H 𝜈 𝑖 𝑧 superscript 1 2 𝑧 𝜈 1 superscript subscript 𝑛 0 superscript 1 2 𝑧 2 𝑛 Euler-Gamma 𝑛 3 2 Euler-Gamma 𝑛 𝜈 3 2 {\displaystyle{\displaystyle\mathbf{L}_{\nu}\left(z\right)=-ie^{-\frac{1}{2}% \pi i\nu}\mathbf{H}_{\nu}\left(iz\right)=(\tfrac{1}{2}z)^{\nu+1}\sum_{n=0}^{% \infty}\frac{(\tfrac{1}{2}z)^{2n}}{\Gamma\left(n+\tfrac{3}{2}\right)\Gamma% \left(n+\nu+\tfrac{3}{2}\right)}.}}
    0 references
    DLMF id
    DLMF:11.2.E2
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    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aadec
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    Struve function
    test
    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
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
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    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
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    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
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