DLMF:10.22.E69 (Q3443): Difference between revisions

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Property / constraint
 

ℜ ⁡ ν > - 1 , ℑ ⁡ z > 0 formulae-sequence 𝜈 1 𝑧 0 {\displaystyle{\displaystyle\Re\nu>-1,\Im z>0}}

\realpart@@{\nu}>-1,\imagpart@@{z}>0
Property / constraint: ℜ ⁡ ν > - 1 , ℑ ⁡ z > 0 formulae-sequence 𝜈 1 𝑧 0 {\displaystyle{\displaystyle\Re\nu>-1,\Im z>0}} / rank
 
Normal rank

Revision as of 16:21, 30 December 2019

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DLMF:10.22.E69
No description defined

    Statements

    test
    ∫ 0 ∞ J ν ⁡ ( a ⁢ t ) ⁢ J ν ⁡ ( b ⁢ t ) ⁢ t ⁢ d t t 2 - z 2 = { 1 2 ⁢ π ⁢ i ⁢ J ν ⁡ ( b ⁢ z ) ⁢ H ν ( 1 ) ⁡ ( a ⁢ z ) , a > b 1 2 ⁢ π ⁢ i ⁢ J ν ⁡ ( a ⁢ z ) ⁢ H ν ( 1 ) ⁡ ( b ⁢ z ) , b > a } , superscript subscript 0 Bessel-J 𝜈 𝑎 𝑡 Bessel-J 𝜈 𝑏 𝑡 𝑡 𝑡 superscript 𝑡 2 superscript 𝑧 2 1 2 𝜋 𝑖 Bessel-J 𝜈 𝑏 𝑧 Hankel-H-1-Bessel-third-kind 𝜈 𝑎 𝑧 𝑎 𝑏 1 2 𝜋 𝑖 Bessel-J 𝜈 𝑎 𝑧 Hankel-H-1-Bessel-third-kind 𝜈 𝑏 𝑧 𝑏 𝑎 {\displaystyle{\displaystyle\int_{0}^{\infty}J_{\nu}\left(at\right)J_{\nu}% \left(bt\right)\frac{t\mathrm{d}t}{t^{2}-z^{2}}=\left\{\begin{array}[]{ll}% \frac{1}{2}\pi iJ_{\nu}\left(bz\right){H^{(1)}_{\nu}}\left(az\right),&a>b\\ \frac{1}{2}\pi iJ_{\nu}\left(az\right){H^{(1)}_{\nu}}\left(bz\right),&b>a\end{% array}\right\},}}
    0 references
    DLMF id
    DLMF:10.22.E69
    0 references
    constraint
    ℜ ⁡ ν > - 1 , ℑ ⁡ z > 0 formulae-sequence 𝜈 1 𝑧 0 {\displaystyle{\displaystyle\Re\nu>-1,\Im z>0}}
    0 references
     
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