DLMF:10.22.E56 (Q3430): Difference between revisions

From testwiki
Jump to navigation Jump to search
← Older edit
imported>Admin
imported>Admin
Β 
(8 intermediate revisions by the same user not shown)
Property / constraint
Β 

0 < a < b 0 π‘Ž 𝑏 {\displaystyle{\displaystyle 0<a<b}}

0<a<b
Property / constraint: 0 < a < b 0 π‘Ž 𝑏 {\displaystyle{\displaystyle 0<a<b}} / rank
Β 
Normal rank
Property / constraint
Β 

β„œ ⁑ ( ΞΌ + Ξ½ + 1 ) > β„œ ⁑ Ξ» > - 1 πœ‡ 𝜈 1 πœ† 1 {\displaystyle{\displaystyle\Re(\mu+\nu+1)>\Re\lambda>-1}}

\Re(\mu+\nu+1)>\Re\lambda>-1
Property / constraint: β„œ ⁑ ( ΞΌ + Ξ½ + 1 ) > β„œ ⁑ Ξ» > - 1 πœ‡ 𝜈 1 πœ† 1 {\displaystyle{\displaystyle\Re(\mu+\nu+1)>\Re\lambda>-1}} / rank
Β 
Normal rank
Property / Symbols used
Β 
Property / Symbols used: Bessel function of the first kind / rank
Β 
Normal rank
Property / Symbols used: Bessel function of the first kind / qualifier
Β 
test:

J Ξ½ ⁑ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}

\BesselJ{\NVar{\nu}}@{\NVar{z}}
Property / Symbols used: Bessel function of the first kind / qualifier
Β 
xml-id: C10.S2.E2.m2aapdec
Property / Symbols used
Β 
Property / Symbols used: gamma function / rank
Β 
Normal rank
Property / Symbols used: gamma function / qualifier
Β 
test:

Ξ“ ⁑ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}

\EulerGamma@{\NVar{z}}
Property / Symbols used: gamma function / qualifier
Β 
xml-id: C5.S2.E1.m2aqdec
Property / Symbols used
Β 
Property / Symbols used: Q10770 / rank
Β 
Normal rank
Property / Symbols used: Q10770 / qualifier
Β 
test:

d x π‘₯ {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
Β 
xml-id: C1.S4.SS4.m1abcdec
Property / Symbols used
Β 
Property / Symbols used: Q10771 / rank
Β 
Normal rank
Property / Symbols used: Q10771 / qualifier
Β 
test:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
Β 
xml-id: C1.S4.SS4.m3abcdec
Property / Symbols used
Β 
Property / Symbols used: Q10811 / rank
Β 
Normal rank
Property / Symbols used: Q10811 / qualifier
Β 
test:

β„œ ⁑ absent {\displaystyle{\displaystyle\Re}}

\realpart@@
Property / Symbols used: Q10811 / qualifier
Β 
xml-id: C1.S9.E2.m1aajdec
Property / Symbols used
Β 
Property / Symbols used: $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function / rank
Β 
Normal rank
Property / Symbols used: $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function / qualifier
Β 
test:

𝐅 ⁑ ( a , b ; c ; z ) scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}

\hyperOlverF@{\NVar{a}}{\NVar{b}}{\NVar{c}}{\NVar{z}}
Property / Symbols used: $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function / qualifier
Β 
xml-id: C15.S2.E2.m2abdec
Property / Symbols used
Β 
Property / Symbols used: Q11427 / rank
Β 
Normal rank
Property / Symbols used: Q11427 / qualifier
Β 
test:

ν 𝜈 {\displaystyle{\displaystyle\nu}}

\nu
Property / Symbols used: Q11427 / qualifier
Β 
xml-id: C10.S1.XMD7.m1aodec

Latest revision as of 12:24, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:10.22.E56
No description defined

    Statements

    test
    ∫ 0 ∞ J ΞΌ ⁑ ( a ⁒ t ) ⁒ J Ξ½ ⁑ ( b ⁒ t ) t Ξ» ⁒ d t = a ΞΌ ⁒ Ξ“ ⁑ ( 1 2 ⁒ Ξ½ + 1 2 ⁒ ΞΌ - 1 2 ⁒ Ξ» + 1 2 ) 2 Ξ» ⁒ b ΞΌ - Ξ» + 1 ⁒ Ξ“ ⁑ ( 1 2 ⁒ Ξ½ - 1 2 ⁒ ΞΌ + 1 2 ⁒ Ξ» + 1 2 ) ⁒ 𝐅 ⁑ ( 1 2 ⁒ ( ΞΌ + Ξ½ - Ξ» + 1 ) , 1 2 ⁒ ( ΞΌ - Ξ½ - Ξ» + 1 ) ; ΞΌ + 1 ; a 2 b 2 ) , superscript subscript 0 Bessel-J πœ‡ π‘Ž 𝑑 Bessel-J 𝜈 𝑏 𝑑 superscript 𝑑 πœ† 𝑑 superscript π‘Ž πœ‡ Euler-Gamma 1 2 𝜈 1 2 πœ‡ 1 2 πœ† 1 2 superscript 2 πœ† superscript 𝑏 πœ‡ πœ† 1 Euler-Gamma 1 2 𝜈 1 2 πœ‡ 1 2 πœ† 1 2 scaled-hypergeometric-bold-F 1 2 πœ‡ 𝜈 πœ† 1 1 2 πœ‡ 𝜈 πœ† 1 πœ‡ 1 superscript π‘Ž 2 superscript 𝑏 2 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{J_{\mu}\left(at\right)J_{% \nu}\left(bt\right)}{t^{\lambda}}\mathrm{d}t=\frac{a^{\mu}\Gamma\left(\frac{1}% {2}\nu+\frac{1}{2}\mu-\frac{1}{2}\lambda+\frac{1}{2}\right)}{2^{\lambda}b^{\mu% -\lambda+1}\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\lambda+\frac{% 1}{2}\right)}\*\mathbf{F}\left(\tfrac{1}{2}(\mu+\nu-\lambda+1),\tfrac{1}{2}(% \mu-\nu-\lambda+1);\mu+1;\frac{a^{2}}{b^{2}}\right),}}
    0 references
    DLMF id
    DLMF:10.22.E56
    0 references
    constraint
    β„œ ⁑ ( ΞΌ + Ξ½ + 1 ) > β„œ ⁑ Ξ» > - 1 πœ‡ 𝜈 1 πœ† 1 {\displaystyle{\displaystyle\Re(\mu+\nu+1)>\Re\lambda>-1}}
    0 references
    0 < a < b 0 π‘Ž 𝑏 {\displaystyle{\displaystyle 0<a<b}}
    0 references
    β„œ ⁑ ( ΞΌ + Ξ½ + 1 ) > β„œ ⁑ Ξ» > - 1 πœ‡ 𝜈 1 πœ† 1 {\displaystyle{\displaystyle\Re(\mu+\nu+1)>\Re\lambda>-1}}
    0 references
    Symbols used
    Bessel function of the first kind
    test
    J Ξ½ ⁑ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2aapdec
    0 references
    gamma function
    test
    Ξ“ ⁑ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aqdec
    0 references
    Q10770
    test
    d x π‘₯ {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abcdec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abcdec
    0 references
    Q10811
    test
    β„œ ⁑ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aajdec
    0 references
    $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function
    test
    𝐅 ⁑ ( a , b ; c ; z ) scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}
    xml-id
    C15.S2.E2.m2abdec
    0 references
    Q11427
    test
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1aodec
    0 references
     
      Retrieved from "https://lct.wmflabs.org/w/index.php?title=Item:Q3430&oldid=18072"