DLMF:1.14.E34 (Q526)

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DLMF:1.14.E34
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    Statements

    Defining formula
    1 2 ⁢ ( f ⁢ ( u + ) + f ⁢ ( u - ) ) = 1 2 ⁢ π ⁢ i ⁢ lim T → ∞ ⁡ ∫ σ - i ⁢ T σ + i ⁢ T u - s ⁢ ℳ ⁡ f ⁡ ( s ) ⁢ d s . 1 2 𝑓 limit-from 𝑢 𝑓 limit-from 𝑢 1 2 𝜋 𝑖 subscript → 𝑇 subscript superscript 𝜎 𝑖 𝑇 𝜎 𝑖 𝑇 superscript 𝑢 𝑠 Mellin-transform 𝑓 𝑠 𝑠 {\displaystyle{\displaystyle\tfrac{1}{2}(f(u+)+f(u-))=\frac{1}{2\pi i}\lim_{T% \to\infty}\int^{\sigma+iT}_{\sigma-iT}u^{-s}\mathscr{M}\mskip-3.0mu f\mskip 3.% 0mu \left(s\right)\mathrm{d}s.}}
    0 references
    DLMF id
    DLMF:1.14.E34
    0 references
    Symbols used
    Mellin transform
    Defining formula
    ℳ ⁡ ( f ) ⁡ ( s ) Mellin-transform 𝑓 𝑠 {\displaystyle{\displaystyle\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}% \right)}}
    xml-id
    C1.S14.E32.m2abdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2akdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1awdec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2afdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3avdec
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