DLMF:2.5.E43 (Q819): Difference between revisions

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

ℒ ⁡ ( f ) ⁡ ( s ) Laplace-transform 𝑓 𝑠 {\displaystyle{\displaystyle\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}% \right)}}

\Laplacetrans@{\NVar{f}}@{\NVar{s}}
Property / Symbols used: Laplace transform / qualifier
 
xml-id: C1.S14.E17.m2abdec

Revision as of 17:07, 1 January 2020

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DLMF:2.5.E43
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    Statements

    Defining formula
    ℒ ⁡ h ⁡ ( ζ ) = ℳ ⁡ h 1 ⁡ ( 1 ) + ∑ ℜ ⁡ β 0 ≤ ℜ ⁡ z ≤ 1 res ⁡ [ - ζ z - 1 ⁢ Γ ⁡ ( 1 - z ) ⁢ ℳ ⁡ h 2 ⁡ ( z ) ] + ∑ 1 < ℜ ⁡ z < l res ⁡ [ - ζ z - 1 ⁢ Γ ⁡ ( 1 - z ) ⁢ ℳ ⁡ h ⁡ ( z ) ] + 1 2 ⁢ π ⁢ i ⁢ ∫ l - δ - i ⁢ ∞ l - δ + i ⁢ ∞ ζ z - 1 ⁢ Γ ⁡ ( 1 - z ) ⁢ ℳ ⁡ h ⁡ ( z ) ⁢ d z , Laplace-transform ℎ 𝜁 Mellin-transform subscript ℎ 1 1 subscript subscript 𝛽 0 𝑧 1 residue delimited-[] superscript 𝜁 𝑧 1 Euler-Gamma 1 𝑧 Mellin-transform subscript ℎ 2 𝑧 subscript 1 𝑧 𝑙 residue delimited-[] superscript 𝜁 𝑧 1 Euler-Gamma 1 𝑧 Mellin-transform ℎ 𝑧 1 2 𝜋 𝑖 superscript subscript 𝑙 𝛿 𝑖 𝑙 𝛿 𝑖 superscript 𝜁 𝑧 1 Euler-Gamma 1 𝑧 Mellin-transform ℎ 𝑧 𝑧 {\displaystyle{\displaystyle\mathscr{L}\mskip-3.0mu h\mskip 3.0mu \left(\zeta% \right)=\mathscr{M}\mskip-3.0mu h_{1}\mskip 3.0mu \left(1\right)+\sum_{\Re% \beta_{0}\leq\Re z\leq 1}\Residue\left[-\zeta^{z-1}\Gamma\left(1-z\right)% \mathscr{M}\mskip-3.0mu h_{2}\mskip 3.0mu \left(z\right)\right]+\sum\limits_{1% <\Re z<l}\Residue\left[-\zeta^{z-1}\Gamma\left(1-z\right)\mathscr{M}\mskip-3.0% mu h\mskip 3.0mu \left(z\right)\right]+\frac{1}{2\pi i}\int_{l-\delta-i\infty}% ^{l-\delta+i\infty}\zeta^{z-1}\Gamma\left(1-z\right)\mathscr{M}\mskip-3.0mu h% \mskip 3.0mu \left(z\right)\mathrm{d}z,}}
    0 references
    DLMF id
    DLMF:2.5.E43
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2afdec
    0 references
    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
     
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