DLMF:10.43.E19 (Q3633): Difference between revisions

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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.m3andec
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
 
test:

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.m2agdec
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.m1addec
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.m1hdec

Latest revision as of 14:02, 2 January 2020

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DLMF:10.43.E19
No description defined

    Statements

    test
    ∫ 0 ∞ t μ - 1 ⁢ K ν ⁡ ( t ) ⁢ d t = 2 μ - 2 ⁢ Γ ⁡ ( 1 2 ⁢ μ - 1 2 ⁢ ν ) ⁢ Γ ⁡ ( 1 2 ⁢ μ + 1 2 ⁢ ν ) , superscript subscript 0 superscript 𝑡 𝜇 1 modified-Bessel-second-kind 𝜈 𝑡 𝑡 superscript 2 𝜇 2 Euler-Gamma 1 2 𝜇 1 2 𝜈 Euler-Gamma 1 2 𝜇 1 2 𝜈 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{\mu-1}K_{\nu}\left(t\right)% \mathrm{d}t=2^{\mu-2}\Gamma\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu\right)\Gamma% \left(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu\right),}}
    0 references
    DLMF id
    DLMF:10.43.E19
    0 references
    constraint
    | ℜ ⁡ ν | < ℜ ⁡ μ 𝜈 𝜇 {\displaystyle{\displaystyle|\Re\nu|<\Re\mu}}
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aedec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1andec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3andec
    0 references
    modified Bessel function of the second kind
    test
    K ν ⁡ ( z ) modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E3.m2agdec
    0 references
    Q10811
    test
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1addec
    0 references
    Q11427
    test
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1hdec
    0 references
     
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