DLMF:2.6.E46 (Q872): 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
 
Defining formula:

γ {\displaystyle{\displaystyle\gamma}}

\EulerConstant
Property / Symbols used: Euler’s constant / qualifier
 
xml-id: C5.S2.E3.m2abdec
Property / Symbols used
 
Property / Symbols used: derivative of $$f$$ with respect to $$x$$ / rank
 
Normal rank
Property / Symbols used: derivative of $$f$$ with respect to $$x$$ / qualifier
 
Defining formula:

d f d x derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}}}

\deriv{\NVar{f}}{\NVar{x}}
Property / Symbols used: derivative of $$f$$ with respect to $$x$$ / qualifier
 
xml-id: C1.S4.E4.m2adec
Property / Symbols used
 
Property / Symbols used: psi (or digamma) function / rank
 
Normal rank
Property / Symbols used: psi (or digamma) function / qualifier
 
Defining formula:

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
 
Defining formula:

! {\displaystyle{\displaystyle!}}

!
Property / Symbols used: Q10755 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r15.m5aidec
Property / Symbols used
 
Property / Symbols used: principal branch of logarithm function / rank
 
Normal rank
Property / Symbols used: principal branch of logarithm function / qualifier
 
Defining formula:

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.m2aedec
Property / Symbols used
 
Property / Symbols used: Q11025 / rank
 
Normal rank
Property / Symbols used: Q11025 / qualifier
 
Defining formula:

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

d_{s}
Property / Symbols used: Q11025 / qualifier
 
xml-id: C2.S6.XMD10.m1ddec
Property / Symbols used
 
Property / Symbols used: Q11028 / rank
 
Normal rank
Property / Symbols used: Q11028 / qualifier
 
Defining formula:

μ 𝜇 {\displaystyle{\displaystyle\mu}}

\mu
Property / Symbols used: Q11028 / qualifier
 
xml-id: C2.S6.XMD14.m1idec
Property / Symbols used
 
Property / Symbols used: fractional integral (locally) / rank
 
Normal rank
Property / Symbols used: fractional integral (locally) / qualifier
 
Defining formula:

I μ superscript 𝐼 𝜇 {\displaystyle{\displaystyle I^{\mu}}}

I^{\mu}
Property / Symbols used: fractional integral (locally) / qualifier
 
xml-id: C2.S6.XMD15.m1cdec
Property / Symbols used
 
Property / Symbols used: Q11019 / rank
 
Normal rank
Property / Symbols used: Q11019 / qualifier
 
Defining formula:

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

f(t)
Property / Symbols used: Q11019 / qualifier
 
xml-id: C2.S6.XMD2.m1wdec
Property / Symbols used
 
Property / Symbols used: Q11031 / rank
 
Normal rank
Property / Symbols used: Q11031 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11031 / qualifier
 
xml-id: C2.S6.XMD20.m1fdec
Property / Symbols used
 
Property / Symbols used: Q11034 / rank
 
Normal rank
Property / Symbols used: Q11034 / qualifier
 
Defining formula:

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

a_{n}
Property / Symbols used: Q11034 / qualifier
 
xml-id: C2.S6.XMD24.m1edec
Property / Symbols used
 
Property / Symbols used: Q11036 / rank
 
Normal rank
Property / Symbols used: Q11036 / qualifier
 
Defining formula:

δ n ⁢ ( x ) subscript 𝛿 𝑛 𝑥 {\displaystyle{\displaystyle\delta_{n}(x)}}

\delta_{n}(x)
Property / Symbols used: Q11036 / qualifier
 
xml-id: C2.S6.XMD25.m1adec

Latest revision as of 17:17, 1 January 2020

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DLMF:2.6.E46
No description defined

    Statements

    Defining formula
    I μ ⁢ f ⁢ ( x ) = ∑ s = 0 n - 1 ( - 1 ) s ⁢ a s s ! ⁢ Γ ⁡ ( μ + 1 ) ⁢ d s + 1 d x s + 1 ⁡ ( x μ ⁢ ( ln ⁡ x - γ - ψ ⁡ ( μ + 1 ) ) ) - ∑ s = 1 n d s Γ ⁡ ( μ - s + 1 ) ⁢ x μ - s + 1 x n ⁢ δ n ⁢ ( x ) , superscript 𝐼 𝜇 𝑓 𝑥 superscript subscript 𝑠 0 𝑛 1 superscript 1 𝑠 subscript 𝑎 𝑠 𝑠 Euler-Gamma 𝜇 1 derivative 𝑥 𝑠 1 superscript 𝑥 𝜇 𝑥 digamma 𝜇 1 superscript subscript 𝑠 1 𝑛 subscript 𝑑 𝑠 Euler-Gamma 𝜇 𝑠 1 superscript 𝑥 𝜇 𝑠 1 superscript 𝑥 𝑛 subscript 𝛿 𝑛 𝑥 {\displaystyle{\displaystyle I^{\mu}f(x)=\sum_{s=0}^{n-1}\frac{(-1)^{s}a_{s}}{% s!\Gamma\left(\mu+1\right)}\frac{{\mathrm{d}}^{s+1}}{{\mathrm{d}x}^{s+1}}\left% (x^{\mu}\left(\ln x-\gamma-\psi\left(\mu+1\right)\right)\right)-\sum_{s=1}^{n}% \frac{d_{s}}{\Gamma\left(\mu-s+1\right)}x^{\mu-s}+\frac{1}{x^{n}}\delta_{n}(x)% ,}}
    0 references
    DLMF id
    DLMF:2.6.E46
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2ahdec
    0 references
    Euler’s constant
    Defining formula
    γ {\displaystyle{\displaystyle\gamma}}
    xml-id
    C5.S2.E3.m2abdec
    0 references
    derivative of $$f$$ with respect to $$x$$
    Defining formula
    d f d x derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}}}
    xml-id
    C1.S4.E4.m2adec
    0 references
    psi (or digamma) function
    Defining formula
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2abdec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aidec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2aedec
    0 references
    Q11025
    Defining formula
    d s subscript 𝑑 𝑠 {\displaystyle{\displaystyle d_{s}}}
    xml-id
    C2.S6.XMD10.m1ddec
    0 references
    Q11028
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C2.S6.XMD14.m1idec
    0 references
    fractional integral (locally)
    Defining formula
    I μ superscript 𝐼 𝜇 {\displaystyle{\displaystyle I^{\mu}}}
    xml-id
    C2.S6.XMD15.m1cdec
    0 references
    Q11019
    Defining formula
    f ⁢ ( t ) 𝑓 𝑡 {\displaystyle{\displaystyle f(t)}}
    xml-id
    C2.S6.XMD2.m1wdec
    0 references
    Q11031
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C2.S6.XMD20.m1fdec
    0 references
    Q11034
    Defining formula
    a n subscript 𝑎 𝑛 {\displaystyle{\displaystyle a_{n}}}
    xml-id
    C2.S6.XMD24.m1edec
    0 references
    Q11036
    Defining formula
    δ n ⁢ ( x ) subscript 𝛿 𝑛 𝑥 {\displaystyle{\displaystyle\delta_{n}(x)}}
    xml-id
    C2.S6.XMD25.m1adec
    0 references
     
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