DLMF:1.14.E20 (Q512): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Q10894 / rank
 
Normal rank
Property / Symbols used: Q10894 / qualifier
 
Defining formula:

σ ∈ ( a , b ) 𝜎 𝑎 𝑏 {\displaystyle{\displaystyle\sigma\in(a,b)}}

\sigma\in(a,b)
Property / Symbols used: Q10894 / qualifier
 
xml-id: C1.S14.XMD11.m1dec

Revision as of 16:24, 1 January 2020

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DLMF:1.14.E20
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    Statements

    Defining formula
    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
    0 references
    constraint
    σ > α 𝜎 𝛼 {\displaystyle{\displaystyle\sigma>\alpha}}
    0 references
    Symbols used
    Laplace transform
    Defining formula
    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
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ajdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aqdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2afdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2addec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3apdec
    0 references
    Q10894
    Defining formula
    σ ∈ ( a , b ) 𝜎 𝑎 𝑏 {\displaystyle{\displaystyle\sigma\in(a,b)}}
    xml-id
    C1.S14.XMD11.m1dec
    0 references
     
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