DLMF:2.6.E44 (Q870): Difference between revisions

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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: 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.m2aadec
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.m5ahdec
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.m2addec
Property / Symbols used
 
Property / Symbols used: Q11025 / rank
 
Normal rank
Property / Symbols used: Q11025 / qualifier
 
test:

d s subscript 𝑑 𝑠 {\displaystyle{\displaystyle d_{s}}}

d_{s}
Property / Symbols used: Q11025 / qualifier
 
xml-id: C2.S6.XMD10.m1cdec
Property / Symbols used
 
Property / Symbols used: Q11028 / rank
 
Normal rank
Property / Symbols used: Q11028 / qualifier
 
test:

μ 𝜇 {\displaystyle{\displaystyle\mu}}

\mu
Property / Symbols used: Q11028 / qualifier
 
xml-id: C2.S6.XMD14.m1gdec
Property / Symbols used
 
Property / Symbols used: convolution (locally) / rank
 
Normal rank
Property / Symbols used: convolution (locally) / qualifier
 
test:

∗ ∗ {\displaystyle{\displaystyle\ast}}

\ast
Property / Symbols used: convolution (locally) / qualifier
 
xml-id: C2.S6.XMD17.m1hdec
Property / Symbols used
 
Property / Symbols used: Q11030 / rank
 
Normal rank
Property / Symbols used: Q11030 / qualifier
 
test:

D 𝐷 {\displaystyle{\displaystyle D}}

D
Property / Symbols used: Q11030 / qualifier
 
xml-id: C2.S6.XMD19.m1bdec
Property / Symbols used
 
Property / Symbols used: Q11019 / rank
 
Normal rank
Property / Symbols used: Q11019 / qualifier
 
test:

f ⁢ ( t ) 𝑓 𝑡 {\displaystyle{\displaystyle f(t)}}

f(t)
Property / Symbols used: Q11019 / qualifier
 
xml-id: C2.S6.XMD2.m1udec
Property / Symbols used
 
Property / Symbols used: Q11031 / rank
 
Normal rank
Property / Symbols used: Q11031 / qualifier
 
test:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11031 / qualifier
 
xml-id: C2.S6.XMD20.m1ddec
Property / Symbols used
 
Property / Symbols used: Q11034 / rank
 
Normal rank
Property / Symbols used: Q11034 / qualifier
 
test:

a n subscript 𝑎 𝑛 {\displaystyle{\displaystyle a_{n}}}

a_{n}
Property / Symbols used: Q11034 / qualifier
 
xml-id: C2.S6.XMD24.m1cdec
Property / Symbols used
 
Property / Symbols used: Q11022 / rank
 
Normal rank
Property / Symbols used: Q11022 / qualifier
 
test:

f n , n ⁢ ( t ) subscript 𝑓 𝑛 𝑛 𝑡 {\displaystyle{\displaystyle f_{n,n}(t)}}

f_{n,n}(t)
Property / Symbols used: Q11022 / qualifier
 
xml-id: C2.S6.XMD7.m1jdec

Latest revision as of 16:16, 1 January 2020

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DLMF:2.6.E44
No description defined

    Statements

    test
    t μ - 1 ∗ f = ∑ s = 0 n - 1 ( - 1 ) s ⁢ a s μ ⋅ s ! ⁢ D s + 1 ⁢ ( t μ ⁢ ( ln ⁡ t - γ - ψ ⁡ ( μ + 1 ) ) ) - ∑ s = 1 n d s ⁢ Γ ⁡ ( μ ) Γ ⁡ ( μ - s + 1 ) ⁢ t μ - s + t μ - 1 ∗ f n ∗ superscript 𝑡 𝜇 1 𝑓 superscript subscript 𝑠 0 𝑛 1 superscript 1 𝑠 subscript 𝑎 𝑠 ⋅ 𝜇 𝑠 superscript 𝐷 𝑠 1 superscript 𝑡 𝜇 𝑡 digamma 𝜇 1 superscript subscript 𝑠 1 𝑛 subscript 𝑑 𝑠 Euler-Gamma 𝜇 Euler-Gamma 𝜇 𝑠 1 superscript 𝑡 𝜇 𝑠 ∗ superscript 𝑡 𝜇 1 subscript 𝑓 𝑛 {\displaystyle{\displaystyle t^{\mu-1}\ast f=\sum_{s=0}^{n-1}\frac{(-1)^{s}a_{% s}}{\mu\cdot s!}D^{s+1}\left(t^{\mu}\left(\ln t-\gamma-\psi\left(\mu+1\right)% \right)\right)-\sum_{s=1}^{n}d_{s}\frac{\Gamma\left(\mu\right)}{\Gamma\left(% \mu-s+1\right)}t^{\mu-s}+t^{\mu-1}\ast f_{n}}}
    0 references
    DLMF id
    DLMF:2.6.E44
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2afdec
    0 references
    Euler’s constant
    test
    γ {\displaystyle{\displaystyle\gamma}}
    xml-id
    C5.S2.E3.m2aadec
    0 references
    psi (or digamma) function
    test
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2aadec
    0 references
    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5ahdec
    0 references
    principal branch of logarithm function
    test
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2addec
    0 references
    Q11025
    test
    d s subscript 𝑑 𝑠 {\displaystyle{\displaystyle d_{s}}}
    xml-id
    C2.S6.XMD10.m1cdec
    0 references
    Q11028
    test
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C2.S6.XMD14.m1gdec
    0 references
    convolution (locally)
    test
    ∗ ∗ {\displaystyle{\displaystyle\ast}}
    xml-id
    C2.S6.XMD17.m1hdec
    0 references
    Q11030
    test
    D 𝐷 {\displaystyle{\displaystyle D}}
    xml-id
    C2.S6.XMD19.m1bdec
    0 references
    Q11019
    test
    f ⁢ ( t ) 𝑓 𝑡 {\displaystyle{\displaystyle f(t)}}
    xml-id
    C2.S6.XMD2.m1udec
    0 references
    Q11031
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C2.S6.XMD20.m1ddec
    0 references
    Q11034
    test
    a n subscript 𝑎 𝑛 {\displaystyle{\displaystyle a_{n}}}
    xml-id
    C2.S6.XMD24.m1cdec
    0 references
    Q11022
    test
    f n , n ⁢ ( t ) subscript 𝑓 𝑛 𝑛 𝑡 {\displaystyle{\displaystyle f_{n,n}(t)}}
    xml-id
    C2.S6.XMD7.m1jdec
    0 references
     
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