DLMF:12.10.E37 (Q4202): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11526 / rank
 
Normal rank
Property / Symbols used: Q11526 / qualifier
 
Defining formula:

s 𝑠 {\displaystyle{\displaystyle s}}

s
Property / Symbols used: Q11526 / qualifier
 
xml-id: C12.S1.XMD5.m1zdec
Property / Symbols used
 
Property / Symbols used: Q11534 / rank
 
Normal rank
Property / Symbols used: Q11534 / qualifier
 
Defining formula:

ζ 𝜁 {\displaystyle{\displaystyle\zeta}}

\zeta
Property / Symbols used: Q11534 / qualifier
 
xml-id: C12.S10.XMD21.m1bdec
Property / Symbols used
 
Property / Symbols used: function (locally) / rank
 
Normal rank
Property / Symbols used: function (locally) / qualifier
 
Defining formula:

ϕ ⁢ ( ζ ) italic-ϕ 𝜁 {\displaystyle{\displaystyle\phi(\zeta)}}

\phi(\zeta)
Property / Symbols used: function (locally) / qualifier
 
xml-id: C12.S10.XMD22.m1bdec
Property / Symbols used
 
Property / Symbols used: Q11535 / rank
 
Normal rank
Property / Symbols used: Q11535 / qualifier
 
Defining formula:

A s ⁢ ( ζ ) subscript 𝐴 𝑠 𝜁 {\displaystyle{\displaystyle A_{s}(\zeta)}}

A_{s}(\zeta)
Property / Symbols used: Q11535 / qualifier
 
xml-id: C12.S10.XMD23.m1adec
Property / Symbols used
 
Property / Symbols used: Q11536 / rank
 
Normal rank
Property / Symbols used: Q11536 / qualifier
 
Defining formula:

B s ⁢ ( ζ ) subscript 𝐵 𝑠 𝜁 {\displaystyle{\displaystyle B_{s}(\zeta)}}

B_{s}(\zeta)
Property / Symbols used: Q11536 / qualifier
 
xml-id: C12.S10.XMD24.m1adec
Property / Symbols used
 
Property / Symbols used: expansion (locally) / rank
 
Normal rank
Property / Symbols used: expansion (locally) / qualifier
 
Defining formula:

g ⁢ ( μ ) 𝑔 𝜇 {\displaystyle{\displaystyle g(\mu)}}

g(\mu)
Property / Symbols used: expansion (locally) / qualifier
 
xml-id: C12.S10.XMD7.m1kdec

Latest revision as of 15:22, 2 January 2020

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DLMF:12.10.E37
No description defined

    Statements

    Defining formula
    V ⁡ ( - 1 2 ⁢ μ 2 , μ ⁢ t ⁢ 2 ) ∼ 2 ⁢ π 1 2 ⁢ μ 1 3 ⁢ g ⁢ ( μ ) ⁢ ϕ ⁢ ( ζ ) Γ ⁡ ( 1 2 + 1 2 ⁢ μ 2 ) ⁢ ( Bi ⁡ ( μ 4 3 ⁢ ζ ) ⁢ ∑ s = 0 ∞ A s ⁢ ( ζ ) μ 4 ⁢ s + Bi ′ ⁡ ( μ 4 3 ⁢ ζ ) μ 8 3 ⁢ ∑ s = 0 ∞ B s ⁢ ( ζ ) μ 4 ⁢ s ) , asymptotic-expansion parabolic-V 1 2 superscript 𝜇 2 𝜇 𝑡 2 2 superscript 𝜋 1 2 superscript 𝜇 1 3 𝑔 𝜇 italic-ϕ 𝜁 Euler-Gamma 1 2 1 2 superscript 𝜇 2 Airy-Bi superscript 𝜇 4 3 𝜁 superscript subscript 𝑠 0 subscript 𝐴 𝑠 𝜁 superscript 𝜇 4 𝑠 diffop Airy-Bi 1 superscript 𝜇 4 3 𝜁 superscript 𝜇 8 3 superscript subscript 𝑠 0 subscript 𝐵 𝑠 𝜁 superscript 𝜇 4 𝑠 {\displaystyle{\displaystyle V\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)% \sim\frac{2\pi^{\frac{1}{2}}\mu^{\frac{1}{3}}g(\mu)\phi(\zeta)}{\Gamma\left(% \tfrac{1}{2}+\tfrac{1}{2}\mu^{2}\right)}\*\left(\mathrm{Bi}\left(\mu^{\frac{4}% {3}}\zeta\right)\sum_{s=0}^{\infty}\frac{A_{s}(\zeta)}{\mu^{4s}}+\frac{\mathrm% {Bi}'\left(\mu^{\frac{4}{3}}\zeta\right)}{\mu^{\frac{8}{3}}}\sum_{s=0}^{\infty% }\frac{B_{s}(\zeta)}{\mu^{4s}}\right),}}
    0 references
    DLMF id
    DLMF:12.10.E37
    0 references
    Symbols used
    Q11081
    Defining formula
    Bi ⁡ ( z ) Airy-Bi 𝑧 {\displaystyle{\displaystyle\mathrm{Bi}\left(\NVar{z}\right)}}
    xml-id
    C9.S2.SS1.m2adec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aedec
    0 references
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11apdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
    0 references
    Q11517
    Defining formula
    V ⁡ ( a , z ) parabolic-V 𝑎 𝑧 {\displaystyle{\displaystyle V\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C12.S2.SS1.p1.m2addec
    0 references
    Q11526
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C12.S1.XMD5.m1zdec
    0 references
    Q11534
    Defining formula
    ζ 𝜁 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C12.S10.XMD21.m1bdec
    0 references
    function (locally)
    Defining formula
    ϕ ⁢ ( ζ ) italic-ϕ 𝜁 {\displaystyle{\displaystyle\phi(\zeta)}}
    xml-id
    C12.S10.XMD22.m1bdec
    0 references
    Q11535
    Defining formula
    A s ⁢ ( ζ ) subscript 𝐴 𝑠 𝜁 {\displaystyle{\displaystyle A_{s}(\zeta)}}
    xml-id
    C12.S10.XMD23.m1adec
    0 references
    Q11536
    Defining formula
    B s ⁢ ( ζ ) subscript 𝐵 𝑠 𝜁 {\displaystyle{\displaystyle B_{s}(\zeta)}}
    xml-id
    C12.S10.XMD24.m1adec
    0 references
    expansion (locally)
    Defining formula
    g ⁢ ( μ ) 𝑔 𝜇 {\displaystyle{\displaystyle g(\mu)}}
    xml-id
    C12.S10.XMD7.m1kdec
    0 references
     
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