DLMF:10.43.E10 (Q3624): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
Defining formula:

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

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1abdec
Property / Symbols used
 
Property / Symbols used: Q11435 / rank
 
Normal rank
Property / Symbols used: Q11435 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11435 / qualifier
 
xml-id: C10.S1.XMD4.m1fdec
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.m1fdec

Latest revision as of 14:00, 2 January 2020

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DLMF:10.43.E10
No description defined

    Statements

    Defining formula
    ∫ x ∞ e t ⁢ t - ν ⁢ K ν ⁡ ( t ) ⁢ d t = e x ⁢ x - ν + 1 2 ⁢ ν - 1 ⁢ ( K ν ⁡ ( x ) + K ν - 1 ⁡ ( x ) ) , superscript subscript 𝑥 superscript 𝑒 𝑡 superscript 𝑡 𝜈 modified-Bessel-second-kind 𝜈 𝑡 𝑡 superscript 𝑒 𝑥 superscript 𝑥 𝜈 1 2 𝜈 1 modified-Bessel-second-kind 𝜈 𝑥 modified-Bessel-second-kind 𝜈 1 𝑥 {\displaystyle{\displaystyle\int_{x}^{\infty}e^{t}t^{-\nu}K_{\nu}\left(t\right% )\mathrm{d}t=\frac{e^{x}x^{-\nu+1}}{2\nu-1}(K_{\nu}\left(x\right)+K_{\nu-1}% \left(x\right)),}}
    0 references
    DLMF id
    DLMF:10.43.E10
    0 references
    constraint
    ℜ ⁡ ν > 1 2 𝜈 1 2 {\displaystyle{\displaystyle\Re\nu>\tfrac{1}{2}}}
    0 references
    Symbols used
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aidec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aedec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aidec
    0 references
    modified Bessel function of the second kind
    Defining formula
    K ν ⁡ ( z ) modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E3.m2abdec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1abdec
    0 references
    Q11435
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C10.S1.XMD4.m1fdec
    0 references
    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1fdec
    0 references
     
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