DLMF:10.22.E43 (Q3417): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: gamma function / rank
 
Normal rank
Property / Symbols used: gamma function / qualifier
 
test:

Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}

\EulerGamma@{\NVar{z}}
Property / Symbols used: gamma function / qualifier
 
xml-id: C5.S2.E1.m2aidec

Revision as of 12:20, 2 January 2020

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DLMF:10.22.E43
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    Statements

    test
    ∫ 0 ∞ t μ ⁢ J ν ⁡ ( t ) ⁢ d t = 2 μ ⁢ Γ ⁡ ( 1 2 ⁢ ν + 1 2 ⁢ μ + 1 2 ) Γ ⁡ ( 1 2 ⁢ ν - 1 2 ⁢ μ + 1 2 ) , superscript subscript 0 superscript 𝑡 𝜇 Bessel-J 𝜈 𝑡 𝑡 superscript 2 𝜇 Euler-Gamma 1 2 𝜈 1 2 𝜇 1 2 Euler-Gamma 1 2 𝜈 1 2 𝜇 1 2 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{\mu}J_{\nu}\left(t\right)% \mathrm{d}t=2^{\mu}\frac{\Gamma\left(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+\tfrac{1}% {2}\right)}{\Gamma\left(\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+\tfrac{1}{2}\right)},}}
    0 references
    DLMF id
    DLMF:10.22.E43
    0 references
    constraint
    ℜ ⁡ ( μ + ν ) > - 1 𝜇 𝜈 1 {\displaystyle{\displaystyle\Re(\mu+\nu)>-1}}
    0 references
    ℜ ⁡ μ < 1 2 𝜇 1 2 {\displaystyle{\displaystyle\Re\mu<\tfrac{1}{2}}}
    0 references
    Symbols used
    Bessel function of the first kind
    test
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2aagdec
    0 references
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aidec
    0 references
     
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