DLMF:10.43.E26 (Q3640): Difference between revisions

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Property / constraint
 

ℜ ⁡ ( ν + 1 - λ ) > | ℜ ⁡ μ | , ℜ ⁡ a > | ℑ ⁡ b | formulae-sequence 𝜈 1 𝜆 𝜇 𝑎 𝑏 {\displaystyle{\displaystyle\Re(\nu+1-\lambda)>|\Re\mu|,\Re a>|\Im b|}}

\realpart@@{(\nu+1-\lambda)}>|\realpart@@{\mu}|,\realpart@@{a}>|\imagpart@@{b}|
Property / constraint: ℜ ⁡ ( ν + 1 - λ ) > | ℜ ⁡ μ | , ℜ ⁡ a > | ℑ ⁡ b | formulae-sequence 𝜈 1 𝜆 𝜇 𝑎 𝑏 {\displaystyle{\displaystyle\Re(\nu+1-\lambda)>|\Re\mu|,\Re a>|\Im b|}} / rank
 
Normal rank

Revision as of 16:40, 30 December 2019

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DLMF:10.43.E26
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    Statements

    test
    ∫ 0 ∞ K μ ⁡ ( a ⁢ t ) ⁢ J ν ⁡ ( b ⁢ t ) t λ ⁢ d t = b ν ⁢ Γ ⁡ ( 1 2 ⁢ ν - 1 2 ⁢ λ + 1 2 ⁢ μ + 1 2 ) ⁢ Γ ⁡ ( 1 2 ⁢ ν - 1 2 ⁢ λ - 1 2 ⁢ μ + 1 2 ) 2 λ + 1 ⁢ a ν - λ + 1 ⁢ 𝐅 ⁡ ( ν - λ + μ + 1 2 , ν - λ - μ + 1 2 ; ν + 1 ; - b 2 a 2 ) , superscript subscript 0 modified-Bessel-second-kind 𝜇 𝑎 𝑡 Bessel-J 𝜈 𝑏 𝑡 superscript 𝑡 𝜆 𝑡 superscript 𝑏 𝜈 Euler-Gamma 1 2 𝜈 1 2 𝜆 1 2 𝜇 1 2 Euler-Gamma 1 2 𝜈 1 2 𝜆 1 2 𝜇 1 2 superscript 2 𝜆 1 superscript 𝑎 𝜈 𝜆 1 scaled-hypergeometric-bold-F 𝜈 𝜆 𝜇 1 2 𝜈 𝜆 𝜇 1 2 𝜈 1 superscript 𝑏 2 superscript 𝑎 2 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{K_{\mu}\left(at\right)J_{% \nu}\left(bt\right)}{t^{\lambda}}\mathrm{d}t=\frac{b^{\nu}\Gamma\left(\frac{1}% {2}\nu-\frac{1}{2}\lambda+\frac{1}{2}\mu+\frac{1}{2}\right)\Gamma\left(\frac{1% }{2}\nu-\frac{1}{2}\lambda-\frac{1}{2}\mu+\frac{1}{2}\right)}{2^{\lambda+1}a^{% \nu-\lambda+1}}\*\mathbf{F}\left(\frac{\nu-\lambda+\mu+1}{2},\frac{\nu-\lambda% -\mu+1}{2};\nu+1;-\frac{b^{2}}{a^{2}}\right),}}
    0 references
    DLMF id
    DLMF:10.43.E26
    0 references
    constraint
    ℜ ⁡ ( ν + 1 - λ ) > | ℜ ⁡ μ | , ℜ ⁡ a > | ℑ ⁡ b | formulae-sequence 𝜈 1 𝜆 𝜇 𝑎 𝑏 {\displaystyle{\displaystyle\Re(\nu+1-\lambda)>|\Re\mu|,\Re a>|\Im b|}}
    0 references
     
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