DLMF:10.32.E2 (Q3522): Difference between revisions

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Property / constraint
 

ℜ ⁡ ν > - 1 2 𝜈 1 2 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2}}}

\realpart@@{\nu}>-\tfrac{1}{2}
Property / constraint: ℜ ⁡ ν > - 1 2 𝜈 1 2 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2}}} / rank
 
Normal rank

Revision as of 17:38, 30 December 2019

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DLMF:10.32.E2
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    Statements

    test
    I ν ⁡ ( z ) = ( 1 2 ⁢ z ) ν π 1 2 ⁢ Γ ⁡ ( ν + 1 2 ) ⁢ ∫ 0 π e ± z ⁢ cos ⁡ θ ⁢ ( sin ⁡ θ ) 2 ⁢ ν ⁢ d θ = ( 1 2 ⁢ z ) ν π 1 2 ⁢ Γ ⁡ ( ν + 1 2 ) ⁢ ∫ - 1 1 ( 1 - t 2 ) ν - 1 2 ⁢ e ± z ⁢ t ⁢ d t , modified-Bessel-first-kind 𝜈 𝑧 superscript 1 2 𝑧 𝜈 superscript 𝜋 1 2 Euler-Gamma 𝜈 1 2 superscript subscript 0 𝜋 superscript 𝑒 plus-or-minus 𝑧 𝜃 superscript 𝜃 2 𝜈 𝜃 superscript 1 2 𝑧 𝜈 superscript 𝜋 1 2 Euler-Gamma 𝜈 1 2 superscript subscript 1 1 superscript 1 superscript 𝑡 2 𝜈 1 2 superscript 𝑒 plus-or-minus 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle I_{\nu}\left(z\right)=\frac{(\frac{1}{2}z)^{\nu}}% {\pi^{\frac{1}{2}}\Gamma\left(\nu+\frac{1}{2}\right)}\int_{0}^{\pi}e^{\pm z% \cos\theta}(\sin\theta)^{2\nu}\mathrm{d}\theta=\frac{(\frac{1}{2}z)^{\nu}}{\pi% ^{\frac{1}{2}}\Gamma\left(\nu+\frac{1}{2}\right)}\int_{-1}^{1}(1-t^{2})^{\nu-% \frac{1}{2}}e^{\pm zt}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:10.32.E2
    0 references
    constraint
    ℜ ⁡ ν > - 1 2 𝜈 1 2 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2}}}
    0 references
     
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