DLMF:2.5.E41 (Q817): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: gamma function / rank
 
Normal rank
Property / Symbols used: gamma function / qualifier
 
Defining formula:

Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}

\EulerGamma@{\NVar{z}}
Property / Symbols used: gamma function / qualifier
 
xml-id: C5.S2.E1.m2addec
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.m2aodec
Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
Defining formula:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2aidec
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
Defining formula:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1aldec
Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
Defining formula:

i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2aidec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
Defining formula:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3aldec
Property / Symbols used
 
Property / Symbols used: Q11005 / rank
 
Normal rank
Property / Symbols used: Q11005 / qualifier
 
Defining formula:

h ⁢ ( x ) ℎ 𝑥 {\displaystyle{\displaystyle h(x)}}

h(x)
Property / Symbols used: Q11005 / qualifier
 
xml-id: C2.S5.XMD16.m1mdec
Property / Symbols used
 
Property / Symbols used: Q11014 / rank
 
Normal rank
Property / Symbols used: Q11014 / qualifier
 
Defining formula:

I j ⁢ ( x ) subscript 𝐼 𝑗 𝑥 {\displaystyle{\displaystyle I_{j}(x)}}

I_{j}(x)
Property / Symbols used: Q11014 / qualifier
 
xml-id: C2.S5.XMD25.m1cdec
Property / Symbols used
 
Property / Symbols used: Q11016 / rank
 
Normal rank
Property / Symbols used: Q11016 / qualifier
 
Defining formula:

ρ 𝜌 {\displaystyle{\displaystyle\rho}}

\rho
Property / Symbols used: Q11016 / qualifier
 
xml-id: C2.S5.XMD27.m1dec

Latest revision as of 17:07, 1 January 2020

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DLMF:2.5.E41
No description defined

    Statements

    Defining formula
    I 1 ⁢ ( x ) = ℳ ⁡ h 1 ⁡ ( 1 ) ⁢ x - 1 + 1 2 ⁢ π ⁢ i ⁢ ∫ ρ - i ⁢ ∞ ρ + i ⁢ ∞ x - z ⁢ Γ ⁡ ( 1 - z ) ⁢ ℳ ⁡ h 1 ⁡ ( z ) ⁢ d z , subscript 𝐼 1 𝑥 Mellin-transform subscript ℎ 1 1 superscript 𝑥 1 1 2 𝜋 𝑖 superscript subscript 𝜌 𝑖 𝜌 𝑖 superscript 𝑥 𝑧 Euler-Gamma 1 𝑧 Mellin-transform subscript ℎ 1 𝑧 𝑧 {\displaystyle{\displaystyle I_{1}(x)=\mathscr{M}\mskip-3.0mu h_{1}\mskip 3.0% mu \left(1\right)x^{-1}+\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_{1}\mskip 3.0mu \left(z% \right)\mathrm{d}z,}}
    0 references
    DLMF id
    DLMF:2.5.E41
    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.m2addec
    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.m2aodec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aidec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aldec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aidec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aldec
    0 references
    Q11005
    Defining formula
    h ⁢ ( x ) ℎ 𝑥 {\displaystyle{\displaystyle h(x)}}
    xml-id
    C2.S5.XMD16.m1mdec
    0 references
    Q11014
    Defining formula
    I j ⁢ ( x ) subscript 𝐼 𝑗 𝑥 {\displaystyle{\displaystyle I_{j}(x)}}
    xml-id
    C2.S5.XMD25.m1cdec
    0 references
    Q11016
    Defining formula
    ρ 𝜌 {\displaystyle{\displaystyle\rho}}
    xml-id
    C2.S5.XMD27.m1dec
    0 references
     
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