DLMF:11.7.E12 (Q4002): 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.m2ahdec

Revision as of 13:50, 2 January 2020

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DLMF:11.7.E12
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    Statements

    test
    ∫ 0 ∞ t - μ - ν ⁢ 𝐇 μ ⁡ ( t ) ⁢ 𝐇 ν ⁡ ( t ) ⁢ d t = π ⁢ Γ ⁡ ( μ + ν ) 2 μ + ν ⁢ Γ ⁡ ( μ + ν + 1 2 ) ⁢ Γ ⁡ ( μ + 1 2 ) ⁢ Γ ⁡ ( ν + 1 2 ) , superscript subscript 0 superscript 𝑡 𝜇 𝜈 Struve-H 𝜇 𝑡 Struve-H 𝜈 𝑡 𝑡 𝜋 Euler-Gamma 𝜇 𝜈 superscript 2 𝜇 𝜈 Euler-Gamma 𝜇 𝜈 1 2 Euler-Gamma 𝜇 1 2 Euler-Gamma 𝜈 1 2 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{-\mu-\nu}\mathbf{H}_{\mu}\left% (t\right)\mathbf{H}_{\nu}\left(t\right)\mathrm{d}t=\frac{\sqrt{\pi}\Gamma\left% (\mu+\nu\right)}{2^{\mu+\nu}\Gamma\left(\mu+\nu+\tfrac{1}{2}\right)\Gamma\left% (\mu+\tfrac{1}{2}\right)\Gamma\left(\nu+\tfrac{1}{2}\right)},}}
    0 references
    DLMF id
    DLMF:11.7.E12
    0 references
    constraint
    ℜ ⁡ ( μ + ν ) > 0 𝜇 𝜈 0 {\displaystyle{\displaystyle\Re(\mu+\nu)>0}}
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2afdec
    0 references
    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.m2aidec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ahdec
    0 references
     
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