DLMF:9.10.E22 (Q2907): Difference between revisions

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Property / constraint
 

n = 1 , 2 , … . 𝑛 1 2 … {\displaystyle{\displaystyle n=1,2,\ldots.}}

n=1,2,\ldots.
Property / constraint: n = 1 , 2 , … . 𝑛 1 2 … {\displaystyle{\displaystyle n=1,2,\ldots.}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: Q10919 / rank
 
Normal rank
Property / Symbols used: Q10919 / qualifier
 
Defining formula:

Ai ⁡ ( z ) Airy-Ai 𝑧 {\displaystyle{\displaystyle\mathrm{Ai}\left(\NVar{z}\right)}}

\AiryAi@{\NVar{z}}
Property / Symbols used: Q10919 / qualifier
 
xml-id: C9.S2.SS1.m1aldec
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: 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: cosine function / rank
 
Normal rank
Property / Symbols used: cosine function / qualifier
 
Defining formula:

cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}

\cos@@{\NVar{z}}
Property / Symbols used: cosine function / qualifier
 
xml-id: C4.S14.E2.m2aadec
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.m1audec
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.m3atdec

Latest revision as of 14:48, 2 January 2020

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DLMF:9.10.E22
No description defined

    Statements

    Defining formula
    ∫ 0 ∞ ∫ t ∞ ⋯ ⁢ ∫ t ∞ Ai ⁡ ( - t ) ⁢ ( d t ) n = 2 ⁢ cos ⁡ ( 1 3 ⁢ ( n - 1 ) ⁢ π ) 3 ( n + 2 ) / 3 ⁢ Γ ⁡ ( 1 3 ⁢ n + 2 3 ) , superscript subscript 0 superscript subscript 𝑡 ⋯ superscript subscript 𝑡 Airy-Ai 𝑡 superscript 𝑡 𝑛 2 1 3 𝑛 1 𝜋 superscript 3 𝑛 2 3 Euler-Gamma 1 3 𝑛 2 3 {\displaystyle{\displaystyle\int_{0}^{\infty}\!\!\int_{t}^{\infty}\!\!\!\!% \cdots\int_{t}^{\infty}\mathrm{Ai}\left({-}t\right)(\mathrm{d}t)^{n}=\frac{2% \cos\left(\tfrac{1}{3}(n-1)\pi\right)}{3^{(n+2)/3}\Gamma\left(\tfrac{1}{3}n+% \tfrac{2}{3}\right)},}}
    0 references
    DLMF id
    DLMF:9.10.E22
    0 references
    constraint
    n = 1 , 2 , … . 𝑛 1 2 … {\displaystyle{\displaystyle n=1,2,\ldots.}}
    0 references
    Symbols used
    Q10919
    Defining formula
    Ai ⁡ ( z ) Airy-Ai 𝑧 {\displaystyle{\displaystyle\mathrm{Ai}\left(\NVar{z}\right)}}
    xml-id
    C9.S2.SS1.m1aldec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2addec
    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
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aadec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1audec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3atdec
    0 references
     
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