DLMF:13.21.E24 (Q4632): Difference between revisions

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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.m2aldec
Property / Symbols used
 
Property / Symbols used: Q11083 / rank
 
Normal rank
Property / Symbols used: Q11083 / qualifier
 
Defining formula:

envBi ⁡ ( x ) envelope-Airy-Bi 𝑥 {\displaystyle{\displaystyle\mathrm{envBi}\left(\NVar{x}\right)}}

\envAiryBi@{\NVar{x}}
Property / Symbols used: Q11083 / qualifier
 
xml-id: C2.S8.SS3.p4.m6aadec
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.m2ajdec
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.m2afdec
Property / Symbols used
 
Property / Symbols used: Q11566 / rank
 
Normal rank
Property / Symbols used: Q11566 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11566 / qualifier
 
xml-id: C13.S1.XMD4.m1udec
Property / Symbols used
 
Property / Symbols used: function (locally) / rank
 
Normal rank
Property / Symbols used: function (locally) / qualifier
 
Defining formula:

c ^ ⁢ ( κ , μ ) ^ 𝑐 𝜅 𝜇 {\displaystyle{\displaystyle\widehat{c}(\kappa,\mu)}}

\widehat{c}(\kappa,\mu)
Property / Symbols used: function (locally) / qualifier
 
xml-id: C13.S21.XMD4.m1bdec
Property / Symbols used
 
Property / Symbols used: function (locally) / rank
 
Normal rank
Property / Symbols used: function (locally) / qualifier
 
Defining formula:

Ψ ^ ⁢ ( κ , μ , x ) ^ Ψ 𝜅 𝜇 𝑥 {\displaystyle{\displaystyle\widehat{\Psi}(\kappa,\mu,x)}}

\widehat{\Psi}(\kappa,\mu,x)
Property / Symbols used: function (locally) / qualifier
 
xml-id: C13.S21.XMD6.m1cdec

Latest revision as of 16:21, 2 January 2020

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English
DLMF:13.21.E24
No description defined

    Statements

    Defining formula
    W - κ , μ ⁡ ( x ⁢ e - π ⁢ i ) = e ( κ - 1 6 ) ⁢ π ⁢ i ⁢ c ^ ⁢ ( κ , μ ) ⁢ Ψ ^ ⁢ ( κ , μ , x ) ⁢ ( Ai ⁡ ( κ 2 3 ⁢ ζ ^ ⁢ e - 2 3 ⁢ π ⁢ i ) + envBi ⁡ ( κ 2 3 ⁢ ζ ^ ) ⁢ O ⁡ ( κ - 1 ) ) , Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑥 superscript 𝑒 𝜋 imaginary-unit superscript 𝑒 𝜅 1 6 𝜋 imaginary-unit ^ 𝑐 𝜅 𝜇 ^ Ψ 𝜅 𝜇 𝑥 Airy-Ai superscript 𝜅 2 3 ^ 𝜁 superscript 𝑒 2 3 𝜋 imaginary-unit envelope-Airy-Bi superscript 𝜅 2 3 ^ 𝜁 Big-O superscript 𝜅 1 {\displaystyle{\displaystyle W_{-\kappa,\mu}\left(xe^{-\pi\mathrm{i}}\right)=e% ^{(\kappa-\frac{1}{6})\pi\mathrm{i}}\widehat{c}(\kappa,\mu)\widehat{\Psi}(% \kappa,\mu,x)\left(\mathrm{Ai}\left(\kappa^{\frac{2}{3}}\widehat{\zeta}e^{-% \frac{2}{3}\pi\mathrm{i}}\right)+\mathrm{envBi}\left(\kappa^{\frac{2}{3}}% \widehat{\zeta}\right)O\left(\kappa^{-1}\right)\right),}}
    0 references
    DLMF id
    DLMF:13.21.E24
    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.m1abdec
    0 references
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2aldec
    0 references
    Whittaker confluent hypergeometric function
    Defining formula
    W κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}
    xml-id
    C13.S14.E3.m2ahdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aldec
    0 references
    Q11083
    Defining formula
    envBi ⁡ ( x ) envelope-Airy-Bi 𝑥 {\displaystyle{\displaystyle\mathrm{envBi}\left(\NVar{x}\right)}}
    xml-id
    C2.S8.SS3.p4.m6aadec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2ajdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2afdec
    0 references
    Q11566
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C13.S1.XMD4.m1udec
    0 references
    function (locally)
    Defining formula
    c ^ ⁢ ( κ , μ ) ^ 𝑐 𝜅 𝜇 {\displaystyle{\displaystyle\widehat{c}(\kappa,\mu)}}
    xml-id
    C13.S21.XMD4.m1bdec
    0 references
    function (locally)
    Defining formula
    Ψ ^ ⁢ ( κ , μ , x ) ^ Ψ 𝜅 𝜇 𝑥 {\displaystyle{\displaystyle\widehat{\Psi}(\kappa,\mu,x)}}
    xml-id
    C13.S21.XMD6.m1cdec
    0 references
     
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