DLMF:10.22.E75 (Q3450): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
Defining formula:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3abudec

Revision as of 13:28, 2 January 2020

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DLMF:10.22.E75
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    Statements

    Defining formula
    ∫ 0 ∞ Y ν ⁡ ( a ⁢ t ) ⁢ J ν ⁡ ( b ⁢ t ) ⁢ J ν ⁡ ( c ⁢ t ) ⁢ t 1 + ν ⁢ d t = { - ( a ⁢ b ⁢ c ) ν ⁢ ( - A ) - ν - 1 2 π 1 2 ⁢ 2 ν + 1 ⁢ Γ ⁡ ( 1 2 - ν ) , 0 < a < | b - c | , 0 , | b - c | < a < b + c , ( a ⁢ b ⁢ c ) ν ⁢ ( - A ) - ν - 1 2 π 1 2 ⁢ 2 ν + 1 ⁢ Γ ⁡ ( 1 2 - ν ) , a > b + c . superscript subscript 0 Bessel-Y-Weber 𝜈 𝑎 𝑡 Bessel-J 𝜈 𝑏 𝑡 Bessel-J 𝜈 𝑐 𝑡 superscript 𝑡 1 𝜈 𝑡 cases superscript 𝑎 𝑏 𝑐 𝜈 superscript 𝐴 𝜈 1 2 superscript 𝜋 1 2 superscript 2 𝜈 1 Euler-Gamma 1 2 𝜈 0 𝑎 𝑏 𝑐 0 𝑏 𝑐 𝑎 𝑏 𝑐 superscript 𝑎 𝑏 𝑐 𝜈 superscript 𝐴 𝜈 1 2 superscript 𝜋 1 2 superscript 2 𝜈 1 Euler-Gamma 1 2 𝜈 𝑎 𝑏 𝑐 {\displaystyle{\displaystyle\int_{0}^{\infty}Y_{\nu}\left(at\right)J_{\nu}% \left(bt\right)J_{\nu}\left(ct\right)t^{1+\nu}\mathrm{d}t=\begin{cases}-\dfrac% {(abc)^{\nu}(-A)^{-\nu-\frac{1}{2}}}{\pi^{\frac{1}{2}}2^{\nu+1}\Gamma\left(% \frac{1}{2}-\nu\right)},&0<a<|b-c|,\\ 0,&|b-c|<a<b+c,\\ \dfrac{(abc)^{\nu}(-A)^{-\nu-\frac{1}{2}}}{\pi^{\frac{1}{2}}2^{\nu+1}\Gamma% \left(\frac{1}{2}-\nu\right)},&a>b+c.\end{cases}}}
    0 references
    DLMF id
    DLMF:10.22.E75
    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.m2abgdec
    0 references
    Bessel function of the second kind
    Defining formula
    Y ν ⁡ ( z ) Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle Y_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E3.m2aldec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2awdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aafdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abudec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abudec
    0 references
     
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