DLMF:2.5.E42 (Q818): Difference between revisions

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

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

\Mellintrans@{\NVar{f}}@{\NVar{s}}
Property / Symbols used: Mellin transform / qualifier
 
xml-id: C1.S14.E32.m2apdec

Revision as of 17:07, 1 January 2020

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DLMF:2.5.E42
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    Statements

    Defining formula
    I 2 ⁢ ( x ) = ∑ ℜ ⁡ β 0 ≤ ℜ ⁡ z ≤ 1 res ⁡ [ - x - z ⁢ Γ ⁡ ( 1 - z ) ⁢ ℳ ⁡ h 2 ⁡ ( z ) ] + 1 2 ⁢ π ⁢ i ⁢ ∫ ρ - i ⁢ ∞ ρ + i ⁢ ∞ x - z ⁢ Γ ⁡ ( 1 - z ) ⁢ ℳ ⁡ h 2 ⁡ ( z ) ⁢ d z . subscript 𝐼 2 𝑥 subscript subscript 𝛽 0 𝑧 1 residue delimited-[] superscript 𝑥 𝑧 Euler-Gamma 1 𝑧 Mellin-transform subscript ℎ 2 𝑧 1 2 𝜋 𝑖 superscript subscript 𝜌 𝑖 𝜌 𝑖 superscript 𝑥 𝑧 Euler-Gamma 1 𝑧 Mellin-transform subscript ℎ 2 𝑧 𝑧 {\displaystyle{\displaystyle I_{2}(x)=\sum_{\Re\beta_{0}\leq\Re z\leq 1}% \Residue\left[-x^{-z}\Gamma\left(1-z\right)\mathscr{M}\mskip-3.0mu h_{2}\mskip 3% .0mu \left(z\right)\right]+\frac{1}{2\pi i}\int_{\rho-i\infty}^{\rho+i\infty}x% ^{-z}\Gamma\left(1-z\right)\mathscr{M}\mskip-3.0mu h_{2}\mskip 3.0mu \left(z% \right)\mathrm{d}z.}}
    0 references
    DLMF id
    DLMF:2.5.E42
    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.m2aedec
    0 references
    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.m2apdec
    0 references
     
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