DLMF:2.6.E28 (Q854)

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DLMF:2.6.E28
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    Statements

    Defining formula
    𝒮 ⁡ f ⁡ ( z ) = ln ⁡ z ⁢ ∑ s = 0 n - 1 ( - 1 ) s ⁢ a s z s + 1 + ∑ s = 0 n - 1 ( - 1 ) s ⁢ d ~ s z s + 1 + R n ⁢ ( z ) , Stieltjes-transform 𝑓 𝑧 𝑧 superscript subscript 𝑠 0 𝑛 1 superscript 1 𝑠 subscript 𝑎 𝑠 superscript 𝑧 𝑠 1 superscript subscript 𝑠 0 𝑛 1 superscript 1 𝑠 subscript ~ 𝑑 𝑠 superscript 𝑧 𝑠 1 subscript 𝑅 𝑛 𝑧 {\displaystyle{\displaystyle\mathcal{S}\mskip-3.0mu f\mskip 3.0mu \left(z% \right)=\ln z\sum_{s=0}^{n-1}(-1)^{s}\frac{a_{s}}{z^{s+1}}+\sum_{s=0}^{n-1}(-1% )^{s}\frac{\widetilde{d}_{s}}{z^{s+1}}+R_{n}(z),}}
    0 references
    DLMF id
    DLMF:2.6.E28
    0 references
    Symbols used
    Stieltjes transform
    Defining formula
    𝒮 ⁡ ( f ) ⁡ ( s ) Stieltjes-transform 𝑓 𝑠 {\displaystyle{\displaystyle\mathcal{S}\left(\NVar{f}\right)\left(\NVar{s}% \right)}}
    xml-id
    C1.S14.E47.m2acdec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2abdec
    0 references
    Q11027
    Defining formula
    R n ⁢ ( z ) subscript 𝑅 𝑛 𝑧 {\displaystyle{\displaystyle R_{n}(z)}}
    xml-id
    C2.S6.XMD12.m1adec
    0 references
    Q11019
    Defining formula
    f ⁢ ( t ) 𝑓 𝑡 {\displaystyle{\displaystyle f(t)}}
    xml-id
    C2.S6.XMD2.m1jdec
    0 references
    Q11020
    Defining formula
    a n subscript 𝑎 𝑛 {\displaystyle{\displaystyle a_{n}}}
    xml-id
    C2.S6.XMD3.m1fdec
    0 references
     
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