DLMF:11.5.E7 (Q3974)

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DLMF:11.5.E7
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    Statements

    Defining formula
    I - ν ⁡ ( x ) - 𝐋 ν ⁡ ( x ) = 2 ⁢ ( 1 2 ⁢ x ) ν π ⁢ Γ ⁡ ( ν + 1 2 ) ⁢ ∫ 0 ∞ ( 1 + t 2 ) ν - 1 2 ⁢ sin ⁡ ( x ⁢ t ) ⁢ d t , modified-Bessel-first-kind 𝜈 𝑥 modified-Struve-L 𝜈 𝑥 2 superscript 1 2 𝑥 𝜈 𝜋 Euler-Gamma 𝜈 1 2 superscript subscript 0 superscript 1 superscript 𝑡 2 𝜈 1 2 𝑥 𝑡 𝑡 {\displaystyle{\displaystyle I_{-\nu}\left(x\right)-\mathbf{L}_{\nu}\left(x% \right)=\frac{2(\tfrac{1}{2}x)^{\nu}}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{1}{2}% \right)}\int_{0}^{\infty}(1+t^{2})^{\nu-\frac{1}{2}}\sin\left(xt\right)\mathrm% {d}t,}}
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    DLMF id
    DLMF:11.5.E7
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    constraint
    ℜ ⁡ ν < 1 2 𝜈 1 2 {\displaystyle{\displaystyle\Re\nu<\tfrac{1}{2}}}
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    x > 0 𝑥 0 {\displaystyle{\displaystyle x>0}}
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    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.m2addec
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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.m2afdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1afdec
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