DLMF:10.43.E27 (Q3641): Difference between revisions

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Changed an Item: Add constraint
Changed an Item: Add constraint
 
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Property / Symbols used
 
Property / Symbols used: Q10901 / rank
 
Normal rank
Property / Symbols used: Q10901 / qualifier
 
Defining formula:

absent {\displaystyle{\displaystyle\Im}}

\imagpart@@
Property / Symbols used: Q10901 / qualifier
 
xml-id: C1.S9.E2.m2acdec
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.m3avdec
Property / Symbols used
 
Property / Symbols used: modified Bessel function of the second kind / rank
 
Normal rank
Property / Symbols used: modified Bessel function of the second kind / qualifier
 
Defining formula:

K ν ( z ) modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}

\modBesselK{\NVar{\nu}}@{\NVar{z}}
Property / Symbols used: modified Bessel function of the second kind / qualifier
 
xml-id: C10.S25.E3.m2amdec
Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
Defining formula:

absent {\displaystyle{\displaystyle\Re}}

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1ajdec
Property / Symbols used
 
Property / Symbols used: Q11427 / rank
 
Normal rank
Property / Symbols used: Q11427 / qualifier
 
Defining formula:

ν 𝜈 {\displaystyle{\displaystyle\nu}}

\nu
Property / Symbols used: Q11427 / qualifier
 
xml-id: C10.S1.XMD7.m1ndec

Latest revision as of 14:04, 2 January 2020

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DLMF:10.43.E27
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    Statements

    0 t μ + ν + 1 K μ ( a t ) J ν ( b t ) d t = ( 2 a ) μ ( 2 b ) ν Γ ( μ + ν + 1 ) ( a 2 + b 2 ) μ + ν + 1 , superscript subscript 0 superscript 𝑡 𝜇 𝜈 1 modified-Bessel-second-kind 𝜇 𝑎 𝑡 Bessel-J 𝜈 𝑏 𝑡 𝑡 superscript 2 𝑎 𝜇 superscript 2 𝑏 𝜈 Euler-Gamma 𝜇 𝜈 1 superscript superscript 𝑎 2 superscript 𝑏 2 𝜇 𝜈 1 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{\mu+\nu+1}K_{\mu}\left(at% \right)J_{\nu}\left(bt\right)\mathrm{d}t=\frac{(2a)^{\mu}(2b)^{\nu}\Gamma\left% (\mu+\nu+1\right)}{(a^{2}+b^{2})^{\mu+\nu+1}},}}
    0 references
    DLMF:10.43.E27
    0 references
    ( ν + 1 ) > | μ | , a > | b | formulae-sequence 𝜈 1 𝜇 𝑎 𝑏 {\displaystyle{\displaystyle\Re(\nu+1)>|\Re\mu|,\Re a>|\Im b|}}
    0 references
    J ν ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    C10.S2.E2.m2aadec
    0 references
    Γ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    C5.S2.E1.m2ahdec
    0 references
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    C1.S4.SS4.m1avdec
    0 references
    absent {\displaystyle{\displaystyle\Im}}
    C1.S9.E2.m2acdec
    0 references
    {\displaystyle{\displaystyle\int}}
    C1.S4.SS4.m3avdec
    0 references
    K ν ( z ) modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}
    C10.S25.E3.m2amdec
    0 references
    absent {\displaystyle{\displaystyle\Re}}
    C1.S9.E2.m1ajdec
    0 references
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    C10.S1.XMD7.m1ndec
    0 references