DLMF:13.10.E14 (Q4473): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / constraint
 

x > 0 𝑥 0 {\displaystyle{\displaystyle x>0}}

x>0
Property / constraint: x > 0 𝑥 0 {\displaystyle{\displaystyle x>0}} / rank
 
Normal rank

Revision as of 17:47, 30 December 2019

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DLMF:13.10.E14
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    Statements

    Defining formula
    ∫ 0 ∞ e - t ⁢ t 1 2 ⁢ ν ⁢ 𝐌 ⁡ ( a , b , t ) ⁢ J ν ⁡ ( 2 ⁢ x ⁢ t ) ⁢ d t = x 1 2 ⁢ ν ⁢ e - x Γ ⁡ ( b - a ) ⁢ U ⁡ ( a , a - b + ν + 2 , x ) , superscript subscript 0 superscript 𝑒 𝑡 superscript 𝑡 1 2 𝜈 Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 𝑡 Bessel-J 𝜈 2 𝑥 𝑡 𝑡 superscript 𝑥 1 2 𝜈 superscript 𝑒 𝑥 Euler-Gamma 𝑏 𝑎 Kummer-confluent-hypergeometric-U 𝑎 𝑎 𝑏 𝜈 2 𝑥 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-t}t^{\frac{1}{2}\nu}{\mathbf{% M}}\left(a,b,t\right)J_{\nu}\left(2\sqrt{xt}\right)\mathrm{d}t=\frac{x^{\frac{% 1}{2}\nu}e^{-x}}{\Gamma\left(b-a\right)}U\left(a,a-b+\nu+2,x\right),}}
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    DLMF id
    DLMF:13.10.E14
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    constraint
    - 1 < ℜ ⁡ ν < 2 ⁢ ℜ ⁡ ( b - a ) - 1 2 1 𝜈 2 𝑏 𝑎 1 2 {\displaystyle{\displaystyle-1<\Re\nu<2\Re(b-a)-\tfrac{1}{2}}}
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    x > 0 𝑥 0 {\displaystyle{\displaystyle x>0}}
    0 references
     
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