DLMF:10.22.E54 (Q3428): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: gamma function / rank
 
Normal rank
Property / Symbols used: gamma function / qualifier
 
test:

Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}

\EulerGamma@{\NVar{z}}
Property / Symbols used: gamma function / qualifier
 
xml-id: C5.S2.E1.m2apdec
Property / Symbols used
 
Property / Symbols used: Olver’s confluent hypergeometric function / rank
 
Normal rank
Property / Symbols used: Olver’s confluent hypergeometric function / qualifier
 
test:

𝐌 ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle{\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right% )}}

\OlverconfhyperM@{\NVar{a}}{\NVar{b}}{\NVar{z}}
Property / Symbols used: Olver’s confluent hypergeometric function / qualifier
 
xml-id: C13.S2.E3.m2adec
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
test:

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

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1abadec
Property / Symbols used
 
Property / Symbols used: exponential function / rank
 
Normal rank
Property / Symbols used: exponential function / qualifier
 
test:

exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}

\exp@@{\NVar{z}}
Property / Symbols used: exponential function / qualifier
 
xml-id: C4.S2.E19.m2acdec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
test:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3abadec
Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
test:

ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1aaidec
Property / Symbols used
 
Property / Symbols used: Q11427 / rank
 
Normal rank
Property / Symbols used: Q11427 / qualifier
 
test:

ν 𝜈 {\displaystyle{\displaystyle\nu}}

\nu
Property / Symbols used: Q11427 / qualifier
 
xml-id: C10.S1.XMD7.m1amdec

Latest revision as of 12:23, 2 January 2020

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DLMF:10.22.E54
No description defined

    Statements

    test
    ∫ 0 ∞ J ν ⁡ ( b ⁢ t ) ⁢ exp ⁡ ( - p 2 ⁢ t 2 ) ⁢ t μ - 1 ⁢ d t = ( 1 2 ⁢ b / p ) ν ⁢ Γ ⁡ ( 1 2 ⁢ ν + 1 2 ⁢ μ ) 2 ⁢ p μ ⁢ exp ⁡ ( - b 2 4 ⁢ p 2 ) ⁢ 𝐌 ⁡ ( 1 2 ⁢ ν - 1 2 ⁢ μ + 1 , ν + 1 , b 2 4 ⁢ p 2 ) , superscript subscript 0 Bessel-J 𝜈 𝑏 𝑡 superscript 𝑝 2 superscript 𝑡 2 superscript 𝑡 𝜇 1 𝑡 superscript 1 2 𝑏 𝑝 𝜈 Euler-Gamma 1 2 𝜈 1 2 𝜇 2 superscript 𝑝 𝜇 superscript 𝑏 2 4 superscript 𝑝 2 Kummer-confluent-hypergeometric-bold-M 1 2 𝜈 1 2 𝜇 1 𝜈 1 superscript 𝑏 2 4 superscript 𝑝 2 {\displaystyle{\displaystyle\int_{0}^{\infty}J_{\nu}\left(bt\right)\exp\left(-% p^{2}t^{2}\right)t^{\mu-1}\mathrm{d}t=\frac{(\tfrac{1}{2}b/p)^{\nu}\Gamma\left% (\tfrac{1}{2}\nu+\tfrac{1}{2}\mu\right)}{2p^{\mu}}\exp\left(-\frac{b^{2}}{4p^{% 2}}\right)\*{\mathbf{M}}\left(\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+1,\nu+1,\frac{b^% {2}}{4p^{2}}\right),}}
    0 references
    DLMF id
    DLMF:10.22.E54
    0 references
    constraint
    ℜ ⁡ ( μ + ν ) > 0 𝜇 𝜈 0 {\displaystyle{\displaystyle\Re(\mu+\nu)>0}}
    0 references
    ℜ ⁡ ( p 2 ) > 0 superscript 𝑝 2 0 {\displaystyle{\displaystyle\Re(p^{2})>0}}
    0 references
    Symbols used
    Bessel function of the first kind
    test
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2aandec
    0 references
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2apdec
    0 references
    Olver’s confluent hypergeometric function
    test
    𝐌 ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle{\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right% )}}
    xml-id
    C13.S2.E3.m2adec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abadec
    0 references
    exponential function
    test
    exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}
    xml-id
    C4.S2.E19.m2acdec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abadec
    0 references
    Q10811
    test
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aaidec
    0 references
    Q11427
    test
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1amdec
    0 references
     
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