DLMF:2.6.E45 (Q871)

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DLMF:2.6.E45
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    Statements

    Defining formula
    I μ ⁢ f ⁢ ( x ) = ∑ s = 0 n - 1 a s ⁢ Γ ⁡ ( 1 - s - α ) Γ ⁡ ( μ + 1 - s - α ) ⁢ x μ - s - α - ∑ s = 1 n c s Γ ⁡ ( μ + 1 - s ) ⁢ x μ - s + 1 x n ⁢ δ n ⁢ ( x ) , superscript 𝐼 𝜇 𝑓 𝑥 superscript subscript 𝑠 0 𝑛 1 subscript 𝑎 𝑠 Euler-Gamma 1 𝑠 𝛼 Euler-Gamma 𝜇 1 𝑠 𝛼 superscript 𝑥 𝜇 𝑠 𝛼 superscript subscript 𝑠 1 𝑛 subscript 𝑐 𝑠 Euler-Gamma 𝜇 1 𝑠 superscript 𝑥 𝜇 𝑠 1 superscript 𝑥 𝑛 subscript 𝛿 𝑛 𝑥 {\displaystyle{\displaystyle I^{\mu}f(x)=\sum_{s=0}^{n-1}a_{s}\frac{\Gamma% \left(1-s-\alpha\right)}{\Gamma\left(\mu+1-s-\alpha\right)}x^{\mu-s-\alpha}-% \sum_{s=1}^{n}\frac{c_{s}}{\Gamma\left(\mu+1-s\right)}x^{\mu-s}+\frac{1}{x^{n}% }\delta_{n}(x),}}
    0 references
    DLMF id
    DLMF:2.6.E45
    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.m2agdec
    0 references
    Q11028
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C2.S6.XMD14.m1hdec
    0 references
    fractional integral (locally)
    Defining formula
    I μ superscript 𝐼 𝜇 {\displaystyle{\displaystyle I^{\mu}}}
    xml-id
    C2.S6.XMD15.m1bdec
    0 references
    Q11019
    Defining formula
    f ⁢ ( t ) 𝑓 𝑡 {\displaystyle{\displaystyle f(t)}}
    xml-id
    C2.S6.XMD2.m1vdec
    0 references
    Q11031
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C2.S6.XMD20.m1edec
    0 references
    Q11034
    Defining formula
    a n subscript 𝑎 𝑛 {\displaystyle{\displaystyle a_{n}}}
    xml-id
    C2.S6.XMD24.m1ddec
    0 references
    Q11036
    Defining formula
    δ n ⁢ ( x ) subscript 𝛿 𝑛 𝑥 {\displaystyle{\displaystyle\delta_{n}(x)}}
    xml-id
    C2.S6.XMD25.m1dec
    0 references
    Q11035
    Defining formula
    c s subscript 𝑐 𝑠 {\displaystyle{\displaystyle c_{s}}}
    xml-id
    C2.S6.XMD9.m1bdec
    0 references
     
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