DLMF:36.9.E8 (Q9961): Difference between revisions

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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.m1aadec
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.m2abdec
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.m1agdec
Property / Symbols used
 
Property / Symbols used: hyperbolic umbilic canonical integral function / rank
 
Normal rank
Property / Symbols used: hyperbolic umbilic canonical integral function / qualifier
 
Defining formula:

Ψ ( H ) ⁡ ( 𝐱 ) hyperbolic-umbilic-canonical-integral 𝐱 {\displaystyle{\displaystyle\Psi^{(\mathrm{H})}\left(\NVar{\mathbf{x}}\right)}}

\hyperumbcanonint@{\NVar{\mathbf{x}}}
Property / Symbols used: hyperbolic umbilic canonical integral function / qualifier
 
xml-id: C36.S2.E5.m4adec
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.m3agdec
Property / Symbols used
 
Property / Symbols used: Q12745 / rank
 
Normal rank
Property / Symbols used: Q12745 / qualifier
 
Defining formula:

y 𝑦 {\displaystyle{\displaystyle y}}

y
Property / Symbols used: Q12745 / qualifier
 
xml-id: C36.S1.XMD10.m1ddec
Property / Symbols used
 
Property / Symbols used: Q12746 / rank
 
Normal rank
Property / Symbols used: Q12746 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q12746 / qualifier
 
xml-id: C36.S1.XMD11.m1bdec
Property / Symbols used
 
Property / Symbols used: Q12748 / rank
 
Normal rank
Property / Symbols used: Q12748 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q12748 / qualifier
 
xml-id: C36.S1.XMD9.m1gdec

Latest revision as of 15:13, 2 January 2020

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English
DLMF:36.9.E8
No description defined

    Statements

    Defining formula
    | Ψ ( H ) ⁡ ( x , y , z ) | 2 = 8 ⁢ π 2 ⁢ ( 2 9 ) 1 / 3 ⁢ ∫ - ∞ ∞ ∫ - ∞ ∞ Ai ⁡ ( ( 4 3 ) 1 / 3 ⁢ ( x + z ⁢ v + 3 ⁢ u 2 ) ) ⁢ Ai ⁡ ( ( 4 3 ) 1 / 3 ⁢ ( y + z ⁢ u + 3 ⁢ v 2 ) ) ⁢ d u ⁢ d v . superscript hyperbolic-umbilic-canonical-integral 𝑥 𝑦 𝑧 2 8 superscript 𝜋 2 superscript 2 9 1 3 superscript subscript superscript subscript Airy-Ai superscript 4 3 1 3 𝑥 𝑧 𝑣 3 superscript 𝑢 2 Airy-Ai superscript 4 3 1 3 𝑦 𝑧 𝑢 3 superscript 𝑣 2 𝑢 𝑣 {\displaystyle{\displaystyle\left|\Psi^{(\mathrm{H})}\left(x,y,z\right)\right|% ^{2}=8\pi^{2}\left(\frac{2}{9}\right)^{1/3}\int_{-\infty}^{\infty}\int_{-% \infty}^{\infty}\mathrm{Ai}\left(\left(\frac{4}{3}\right)^{1/3}(x+zv+3u^{2})% \right)\mathrm{Ai}\left(\left(\frac{4}{3}\right)^{1/3}(y+zu+3v^{2})\right)% \mathrm{d}u\mathrm{d}v.}}
    0 references
    DLMF id
    DLMF:36.9.E8
    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.m1aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1agdec
    0 references
    hyperbolic umbilic canonical integral function
    Defining formula
    Ψ ( H ) ⁡ ( 𝐱 ) hyperbolic-umbilic-canonical-integral 𝐱 {\displaystyle{\displaystyle\Psi^{(\mathrm{H})}\left(\NVar{\mathbf{x}}\right)}}
    xml-id
    C36.S2.E5.m4adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3agdec
    0 references
    Q12745
    Defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C36.S1.XMD10.m1ddec
    0 references
    Q12746
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C36.S1.XMD11.m1bdec
    0 references
    Q12748
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C36.S1.XMD9.m1gdec
    0 references
     
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