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

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Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
test:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2aadec

Revision as of 13:40, 2 January 2020

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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
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
     
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