DLMF:1.14.E20 (Q512)

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DLMF:1.14.E20
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    Statements

    test
    f ⁢ ( t ) = 1 2 ⁢ π ⁢ i ⁢ lim T → ∞ ⁡ ∫ σ - i ⁢ T σ + i ⁢ T e t ⁢ s ⁢ ℒ ⁡ f ⁡ ( s ) ⁢ d s , 𝑓 𝑡 1 2 𝜋 𝑖 subscript → 𝑇 subscript superscript 𝜎 𝑖 𝑇 𝜎 𝑖 𝑇 superscript 𝑒 𝑡 𝑠 Laplace-transform 𝑓 𝑠 𝑠 {\displaystyle{\displaystyle f(t)=\frac{1}{2\pi i}\lim_{T\to\infty}\int^{% \sigma+iT}_{\sigma-iT}e^{ts}\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(s% \right)\mathrm{d}s,}}
    0 references
    DLMF id
    DLMF:1.14.E20
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    constraint
    σ > α 𝜎 𝛼 {\displaystyle{\displaystyle\sigma>\alpha}}
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    Symbols used
    Laplace transform
    test
    ℒ ⁡ ( f ) ⁡ ( s ) Laplace-transform 𝑓 𝑠 {\displaystyle{\displaystyle\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}% \right)}}
    xml-id
    C1.S14.E17.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.m2ajdec
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    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aqdec
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    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2afdec
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