DLMF:2.3.E15 (Q732): 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.m2abdec
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.m11ahdec
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.m1aidec
Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
Defining formula:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2agdec
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.m3aidec
Property / Symbols used
 
Property / Symbols used: Q10956 / rank
 
Normal rank
Property / Symbols used: Q10956 / qualifier
 
Defining formula:

a 𝑎 {\displaystyle{\displaystyle a}}

a
Property / Symbols used: Q10956 / qualifier
 
xml-id: C2.S3.XMD15.m1bdec
Property / Symbols used
 
Property / Symbols used: Q10957 / rank
 
Normal rank
Property / Symbols used: Q10957 / qualifier
 
Defining formula:

b 𝑏 {\displaystyle{\displaystyle b}}

b
Property / Symbols used: Q10957 / qualifier
 
xml-id: C2.S3.XMD16.m1adec
Property / Symbols used
 
Property / Symbols used: Q10958 / rank
 
Normal rank
Property / Symbols used: Q10958 / qualifier
 
Defining formula:

p ⁢ ( t ) 𝑝 𝑡 {\displaystyle{\displaystyle p(t)}}

p(t)
Property / Symbols used: Q10958 / qualifier
 
xml-id: C2.S3.XMD17.m1bdec
Property / Symbols used
 
Property / Symbols used: Q10959 / rank
 
Normal rank
Property / Symbols used: Q10959 / qualifier
 
Defining formula:

q ⁢ ( t ) 𝑞 𝑡 {\displaystyle{\displaystyle q(t)}}

q(t)
Property / Symbols used: Q10959 / qualifier
 
xml-id: C2.S3.XMD19.m1bdec
Property / Symbols used
 
Property / Symbols used: Q10950 / rank
 
Normal rank
Property / Symbols used: Q10950 / qualifier
 
Defining formula:

λ 𝜆 {\displaystyle{\displaystyle\lambda}}

\lambda
Property / Symbols used: Q10950 / qualifier
 
xml-id: C2.S3.XMD8.m1edec
Property / Symbols used
 
Property / Symbols used: Q10951 / rank
 
Normal rank
Property / Symbols used: Q10951 / qualifier
 
Defining formula:

μ 𝜇 {\displaystyle{\displaystyle\mu}}

\mu
Property / Symbols used: Q10951 / qualifier
 
xml-id: C2.S3.XMD9.m1edec

Latest revision as of 16:52, 1 January 2020

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DLMF:2.3.E15
No description defined

    Statements

    Defining formula
    ∫ a b e - x ⁢ p ⁢ ( t ) ⁢ q ⁢ ( t ) ⁢ d t ∼ e - x ⁢ p ⁢ ( a ) ⁢ ∑ s = 0 ∞ Γ ⁡ ( s + λ μ ) ⁢ b s x ( s + λ ) / μ , asymptotic-expansion superscript subscript 𝑎 𝑏 superscript 𝑒 𝑥 𝑝 𝑡 𝑞 𝑡 𝑡 superscript 𝑒 𝑥 𝑝 𝑎 superscript subscript 𝑠 0 Euler-Gamma 𝑠 𝜆 𝜇 subscript 𝑏 𝑠 superscript 𝑥 𝑠 𝜆 𝜇 {\displaystyle{\displaystyle\int_{a}^{b}e^{-xp(t)}q(t)\mathrm{d}t\sim e^{-xp(a% )}\sum_{s=0}^{\infty}\Gamma\left(\frac{s+\lambda}{\mu}\right)\frac{b_{s}}{x^{(% s+\lambda)/\mu}},}}
    0 references
    DLMF id
    DLMF:2.3.E15
    0 references
    constraint
    x → + ∞ → 𝑥 {\displaystyle{\displaystyle x\to+\infty}}
    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.m2abdec
    0 references
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11ahdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aidec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2agdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aidec
    0 references
    Q10956
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C2.S3.XMD15.m1bdec
    0 references
    Q10957
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C2.S3.XMD16.m1adec
    0 references
    Q10958
    Defining formula
    p ⁢ ( t ) 𝑝 𝑡 {\displaystyle{\displaystyle p(t)}}
    xml-id
    C2.S3.XMD17.m1bdec
    0 references
    Q10959
    Defining formula
    q ⁢ ( t ) 𝑞 𝑡 {\displaystyle{\displaystyle q(t)}}
    xml-id
    C2.S3.XMD19.m1bdec
    0 references
    Q10950
    Defining formula
    λ 𝜆 {\displaystyle{\displaystyle\lambda}}
    xml-id
    C2.S3.XMD8.m1edec
    0 references
    Q10951
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C2.S3.XMD9.m1edec
    0 references
     
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