DLMF:2.5.E43 (Q819)

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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
    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.m2aqdec
    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
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1andec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2akdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3andec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1agdec
    0 references
    Q10960
    Defining formula
    res residue {\displaystyle{\displaystyle\Residue}}
    xml-id
    C1.S10.SS3.p5.m4addec
    0 references
    Q11005
    Defining formula
    h ⁢ ( x ) ℎ 𝑥 {\displaystyle{\displaystyle h(x)}}
    xml-id
    C2.S5.XMD16.m1odec
    0 references
    Q11017
    Defining formula
    δ 𝛿 {\displaystyle{\displaystyle\delta}}
    xml-id
    C2.S5.XMD28.m1dec
    0 references
     
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