DLMF:10.22.E40 (Q3414): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Bessel function of the second kind / rank
 
Normal rank
Property / Symbols used: Bessel function of the second kind / qualifier
 
test:

Y ν ⁡ ( z ) Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle Y_{\NVar{\nu}}\left(\NVar{z}\right)}}

\BesselY{\NVar{\nu}}@{\NVar{z}}
Property / Symbols used: Bessel function of the second kind / qualifier
 
xml-id: C10.S2.E3.m2acdec
Property / Symbols used
 
Property / Symbols used: Euler’s constant / rank
 
Normal rank
Property / Symbols used: Euler’s constant / qualifier
 
test:

γ {\displaystyle{\displaystyle\gamma}}

\EulerConstant
Property / Symbols used: Euler’s constant / qualifier
 
xml-id: C5.S2.E3.m2aadec
Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
test:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2aodec
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.m1aamdec
Property / Symbols used
 
Property / Symbols used: psi (or digamma) function / rank
 
Normal rank
Property / Symbols used: psi (or digamma) function / qualifier
 
test:

ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}

\digamma@{\NVar{z}}
Property / Symbols used: psi (or digamma) function / qualifier
 
xml-id: C5.S2.E2.m2abdec
Property / Symbols used
 
Property / Symbols used: Q10755 / rank
 
Normal rank
Property / Symbols used: Q10755 / qualifier
 
test:

! {\displaystyle{\displaystyle!}}

!
Property / Symbols used: Q10755 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r15.m5acdec
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.m3aamdec
Property / Symbols used
 
Property / Symbols used: principal branch of logarithm function / rank
 
Normal rank
Property / Symbols used: principal branch of logarithm function / qualifier
 
test:

ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}

\ln@@{\NVar{z}}
Property / Symbols used: principal branch of logarithm function / qualifier
 
xml-id: C4.S2.E2.m2aadec
Property / Symbols used
 
Property / Symbols used: Q11428 / rank
 
Normal rank
Property / Symbols used: Q11428 / qualifier
 
test:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11428 / qualifier
 
xml-id: C10.S1.XMD3.m1jdec

Latest revision as of 12:19, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:10.22.E40
No description defined

    Statements

    test
    ∫ x ∞ Y 0 ⁡ ( t ) t ⁢ d t = - 1 π ⁢ ( ln ⁡ ( 1 2 ⁢ x ) + γ ) 2 + π 6 + 2 π ⁢ ∑ k = 1 ∞ ( - 1 ) k ⁢ ( ψ ⁡ ( k + 1 ) + 1 2 ⁢ k - ln ⁡ ( 1 2 ⁢ x ) ) ⁢ ( 1 2 ⁢ x ) 2 ⁢ k 2 ⁢ k ⁢ ( k ! ) 2 , superscript subscript 𝑥 Bessel-Y-Weber 0 𝑡 𝑡 𝑡 1 𝜋 superscript 1 2 𝑥 2 𝜋 6 2 𝜋 superscript subscript 𝑘 1 superscript 1 𝑘 digamma 𝑘 1 1 2 𝑘 1 2 𝑥 superscript 1 2 𝑥 2 𝑘 2 𝑘 superscript 𝑘 2 {\displaystyle{\displaystyle\int_{x}^{\infty}\frac{Y_{0}\left(t\right)}{t}% \mathrm{d}t=-\frac{1}{\pi}\left(\ln\left(\tfrac{1}{2}x\right)+\gamma\right)^{2% }+\frac{\pi}{6}+\frac{2}{\pi}\sum_{k=1}^{\infty}(-1)^{k}\*\left(\psi\left(k+1% \right)+\frac{1}{2k}-\ln\left(\tfrac{1}{2}x\right)\right)\frac{(\tfrac{1}{2}x)% ^{2k}}{2k(k!)^{2}},}}
    0 references
    DLMF id
    DLMF:10.22.E40
    0 references
    Symbols used
    Bessel function of the second kind
    test
    Y ν ⁡ ( z ) Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle Y_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E3.m2acdec
    0 references
    Euler’s constant
    test
    γ {\displaystyle{\displaystyle\gamma}}
    xml-id
    C5.S2.E3.m2aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aodec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aamdec
    0 references
    psi (or digamma) function
    test
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2abdec
    0 references
    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5acdec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aamdec
    0 references
    principal branch of logarithm function
    test
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2aadec
    0 references
    Q11428
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C10.S1.XMD3.m1jdec
    0 references
     
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