DLMF:36.2.E9 (Q9845): 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.m2acdec
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.m1aedec
Property / Symbols used
 
Property / Symbols used: exponential function / rank
 
Normal rank
Property / Symbols used: exponential function / qualifier
 
Defining formula:

exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}

\exp@@{\NVar{z}}
Property / Symbols used: exponential function / qualifier
 
xml-id: C4.S2.E19.m2aedec
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.m4abdec
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.m2aedec
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.m3aedec
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.m1edec
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.m1edec
Property / Symbols used
 
Property / Symbols used: Q12747 / rank
 
Normal rank
Property / Symbols used: Q12747 / qualifier
 
Defining formula:

s 𝑠 {\displaystyle{\displaystyle s}}

s
Property / Symbols used: Q12747 / qualifier
 
xml-id: C36.S1.XMD6.m1cdec
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.m1edec

Latest revision as of 15:04, 2 January 2020

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DLMF:36.2.E9
No description defined

    Statements

    Defining formula
    Ψ ( H ) ⁡ ( 𝐱 ) = 2 ⁢ π 3 1 / 3 ⁢ ∫ ∞ ⁢ exp ⁡ ( 5 ⁢ π ⁢ i / 6 ) ∞ ⁢ exp ⁡ ( π ⁢ i / 6 ) exp ⁡ ( i ⁢ ( s 3 + x ⁢ s ) ) ⁢ Ai ⁡ ( z ⁢ s + y 3 1 / 3 ) ⁢ d s . hyperbolic-umbilic-canonical-integral 𝐱 2 𝜋 superscript 3 1 3 superscript subscript 5 𝜋 𝑖 6 𝜋 𝑖 6 𝑖 superscript 𝑠 3 𝑥 𝑠 Airy-Ai 𝑧 𝑠 𝑦 superscript 3 1 3 𝑠 {\displaystyle{\displaystyle\Psi^{(\mathrm{H})}\left(\mathbf{x}\right)=\frac{2% \pi}{3^{1/3}}\int_{\infty\exp\left(5\pi i/6\right)}^{\infty\exp\left(\pi i/6% \right)}\exp\left(i(s^{3}+xs)\right)\mathrm{Ai}\left(\frac{zs+y}{3^{1/3}}% \right)\mathrm{d}s.}}
    0 references
    DLMF id
    DLMF:36.2.E9
    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.m2acdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aedec
    0 references
    exponential function
    Defining formula
    exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}
    xml-id
    C4.S2.E19.m2aedec
    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.m4abdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aedec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aedec
    0 references
    Q12745
    Defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C36.S1.XMD10.m1edec
    0 references
    Q12746
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C36.S1.XMD11.m1edec
    0 references
    Q12747
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C36.S1.XMD6.m1cdec
    0 references
    Q12748
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C36.S1.XMD9.m1edec
    0 references
     
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