DLMF:1.14.E49 (Q541)

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DLMF:1.14.E49
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    Statements

    test
    lim t → 0 + ⁡ 𝒮 ⁡ f ⁡ ( - σ - i ⁢ t ) - 𝒮 ⁡ f ⁡ ( - σ + i ⁢ t ) 2 ⁢ π ⁢ i = 1 2 ⁢ ( f ⁢ ( σ + ) + f ⁢ ( σ - ) ) , subscript → 𝑡 limit-from 0 Stieltjes-transform 𝑓 𝜎 𝑖 𝑡 Stieltjes-transform 𝑓 𝜎 𝑖 𝑡 2 𝜋 𝑖 1 2 𝑓 limit-from 𝜎 𝑓 limit-from 𝜎 {\displaystyle{\displaystyle\lim_{t\to 0+}\frac{\mathcal{S}\mskip-3.0mu f% \mskip 3.0mu \left(-\sigma-it\right)-\mathcal{S}\mskip-3.0mu f\mskip 3.0mu % \left(-\sigma+it\right)}{2\pi i}=\tfrac{1}{2}(f(\sigma+)+f(\sigma-)),}}
    0 references
    DLMF id
    DLMF:1.14.E49
    0 references
    Symbols used
    Stieltjes transform
    test
    𝒮 ⁡ ( f ) ⁡ ( s ) Stieltjes-transform 𝑓 𝑠 {\displaystyle{\displaystyle\mathcal{S}\left(\NVar{f}\right)\left(\NVar{s}% \right)}}
    xml-id
    C1.S14.E47.m2abdec
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    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2audec
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    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aldec
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