DLMF:12.12.E3 (Q4228): Difference between revisions

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Property / constraint
 

ℜ ⁡ a < 1 2 , x > 0 formulae-sequence 𝑎 1 2 𝑥 0 {\displaystyle{\displaystyle\Re a<\tfrac{1}{2},x>0}}

\realpart@@{a}<\tfrac{1}{2},x>0
Property / constraint: ℜ ⁡ a < 1 2 , x > 0 formulae-sequence 𝑎 1 2 𝑥 0 {\displaystyle{\displaystyle\Re a<\tfrac{1}{2},x>0}} / rank
 
Normal rank

Revision as of 16:44, 30 December 2019

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DLMF:12.12.E3
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    Statements

    test
    ∫ 0 ∞ e - 1 4 ⁢ t 2 ⁢ t - a - 1 2 ⁢ ( x 2 + t 2 ) - 1 ⁢ U ⁡ ( a , t ) ⁢ d t = π / 2 ⁢ Γ ⁡ ( 1 2 - a ) ⁢ x - a - 3 2 ⁢ e 1 4 ⁢ x 2 ⁢ U ⁡ ( - a , x ) , superscript subscript 0 superscript 𝑒 1 4 superscript 𝑡 2 superscript 𝑡 𝑎 1 2 superscript superscript 𝑥 2 superscript 𝑡 2 1 parabolic-U 𝑎 𝑡 𝑡 𝜋 2 Euler-Gamma 1 2 𝑎 superscript 𝑥 𝑎 3 2 superscript 𝑒 1 4 superscript 𝑥 2 parabolic-U 𝑎 𝑥 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-\frac{1}{4}t^{2}}t^{-a-\frac{% 1}{2}}(x^{2}+t^{2})^{-1}U\left(a,t\right)\mathrm{d}t=\sqrt{\pi/2}\Gamma\left(% \tfrac{1}{2}-a\right)x^{-a-\frac{3}{2}}e^{\frac{1}{4}x^{2}}U\left(-a,x\right),}}
    0 references
    DLMF id
    DLMF:12.12.E3
    0 references
    constraint
    ℜ ⁡ a < 1 2 , x > 0 formulae-sequence 𝑎 1 2 𝑥 0 {\displaystyle{\displaystyle\Re a<\tfrac{1}{2},x>0}}
    0 references
     
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