DLMF:2.5.E44 (Q820): Difference between revisions

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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.m2acdec
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.m2ardec
Property / Symbols used
 
Property / Symbols used: Q10931 / rank
 
Normal rank
Property / Symbols used: Q10931 / qualifier
 
Defining formula:

∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}

\asympexp
Property / Symbols used: Q10931 / qualifier
 
xml-id: C2.S1.SS3.p1.m11abdec
Property / Symbols used
 
Property / Symbols used: Q10755 / rank
 
Normal rank
Property / Symbols used: Q10755 / qualifier
 
Defining formula:

! {\displaystyle{\displaystyle!}}

!
Property / Symbols used: Q10755 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r15.m5adec
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.m1pdec
Property / Symbols used
 
Property / Symbols used: Q11008 / rank
 
Normal rank
Property / Symbols used: Q11008 / qualifier
 
Defining formula:

b 𝑏 {\displaystyle{\displaystyle b}}

b
Property / Symbols used: Q11008 / qualifier
 
xml-id: C2.S5.XMD3.m1edec

Latest revision as of 17:08, 1 January 2020

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DLMF:2.5.E44
No description defined

    Statements

    Defining formula
    ℒ ⁡ h ⁡ ( ζ ) ∼ ∑ n = 0 ∞ b n ⁢ Γ ⁡ ( 1 - β n ) ⁢ ζ β n - 1 + ∑ n = 0 ∞ ( - ζ ) n n ! ⁢ ℳ ⁡ h ⁡ ( n + 1 ) , asymptotic-expansion Laplace-transform ℎ 𝜁 superscript subscript 𝑛 0 subscript 𝑏 𝑛 Euler-Gamma 1 subscript 𝛽 𝑛 superscript 𝜁 subscript 𝛽 𝑛 1 superscript subscript 𝑛 0 superscript 𝜁 𝑛 𝑛 Mellin-transform ℎ 𝑛 1 {\displaystyle{\displaystyle\mathscr{L}\mskip-3.0mu h\mskip 3.0mu \left(\zeta% \right)\sim\sum_{n=0}^{\infty}b_{n}\Gamma\left(1-\beta_{n}\right)\zeta^{\beta_% {n}-1}+\sum\limits_{n=0}^{\infty}\frac{(-\zeta)^{n}}{n!}\mathscr{M}\mskip-3.0% mu h\mskip 3.0mu \left(n+1\right),}}
    0 references
    DLMF id
    DLMF:2.5.E44
    0 references
    constraint
    ζ → 0 + → 𝜁 limit-from 0 {\displaystyle{\displaystyle\zeta\to 0+}}
    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.m2agdec
    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.m2acdec
    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.m2ardec
    0 references
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11abdec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
    0 references
    Q11005
    Defining formula
    h ⁢ ( x ) ℎ 𝑥 {\displaystyle{\displaystyle h(x)}}
    xml-id
    C2.S5.XMD16.m1pdec
    0 references
    Q11008
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C2.S5.XMD3.m1edec
    0 references
     
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