DLMF:13.10.E15 (Q4474): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
Defining formula:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1andec

Revision as of 16:02, 2 January 2020

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DLMF:13.10.E15
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    Statements

    Defining formula
    ∫ 0 ∞ t 1 2 ⁢ ν ⁢ U ⁡ ( a , b , t ) ⁢ J ν ⁡ ( 2 ⁢ x ⁢ t ) ⁢ d t = Γ ⁡ ( ν - b + 2 ) Γ ⁡ ( a ) ⁢ x 1 2 ⁢ ν ⁢ U ⁡ ( ν - b + 2 , ν - a + 2 , x ) , superscript subscript 0 superscript 𝑡 1 2 𝜈 Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑡 Bessel-J 𝜈 2 𝑥 𝑡 𝑡 Euler-Gamma 𝜈 𝑏 2 Euler-Gamma 𝑎 superscript 𝑥 1 2 𝜈 Kummer-confluent-hypergeometric-U 𝜈 𝑏 2 𝜈 𝑎 2 𝑥 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{\frac{1}{2}\nu}U\left(a,b,t% \right)J_{\nu}\left(2\sqrt{xt}\right)\mathrm{d}t=\frac{\Gamma\left(\nu-b+2% \right)}{\Gamma\left(a\right)}x^{\frac{1}{2}\nu}U\left(\nu-b+2,\nu-a+2,x\right% ),}}
    0 references
    DLMF id
    DLMF:13.10.E15
    0 references
    constraint
    max ⁡ ( ℜ ⁡ b - 2 , - 1 ) < ℜ ⁡ ν < 2 ⁢ ℜ ⁡ a + 1 2 𝑏 2 1 𝜈 2 𝑎 1 2 {\displaystyle{\displaystyle\max\left(\Re b-2,-1\right)<\Re\nu<2\Re a+\tfrac{1% }{2}}}
    0 references
    x > 0 𝑥 0 {\displaystyle{\displaystyle x>0}}
    0 references
    max ⁡ ( ℜ ⁡ b - 2 , - 1 ) < ℜ ⁡ ν < 2 ⁢ ℜ ⁡ a + 1 2 𝑏 2 1 𝜈 2 𝑎 1 2 {\displaystyle{\displaystyle\max\left(\Re b-2,-1\right)<\Re\nu<2\Re a+\tfrac{1% }{2}}}
    0 references
    Symbols used
    Bessel function of the first kind
    Defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2abdec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2ajdec
    0 references
    Kummer confluent hypergeometric function
    Defining formula
    U ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{b},\NVar{z}\right)}}
    xml-id
    C13.S2.E6.m2agdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1andec
    0 references
     
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