DLMF:33.12.E2 (Q9596): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10931 / rank
 
Normal rank
Property / Symbols used: Q10931 / qualifier
 
Defining formula:

∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}

\asympexp
Property / Symbols used: Q10931 / qualifier
 
xml-id: C2.S1.SS3.p1.m11adec
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.m2adec
Property / Symbols used
 
Property / Symbols used: Q12670 / rank
 
Normal rank
Property / Symbols used: Q12670 / qualifier
 
Defining formula:

G ℓ ⁡ ( η , ρ ) irregular-Coulomb-G ℓ 𝜂 𝜌 {\displaystyle{\displaystyle G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}% \right)}}

\irregCoulombG{\NVar{\ell}}@{\NVar{\eta}}{\NVar{\rho}}
Property / Symbols used: Q12670 / qualifier
 
xml-id: C33.S2.E11.m1adec
Property / Symbols used
 
Property / Symbols used: regular Coulomb radial function / rank
 
Normal rank
Property / Symbols used: regular Coulomb radial function / qualifier
 
Defining formula:

F ℓ ⁡ ( η , ρ ) regular-Coulomb-F ℓ 𝜂 𝜌 {\displaystyle{\displaystyle F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}% \right)}}

\regCoulombF{\NVar{\ell}}@{\NVar{\eta}}{\NVar{\rho}}
Property / Symbols used: regular Coulomb radial function / qualifier
 
xml-id: C33.S2.E3.m2adec
Property / Symbols used
 
Property / Symbols used: Q12667 / rank
 
Normal rank
Property / Symbols used: Q12667 / qualifier
 
Defining formula:

ρ 𝜌 {\displaystyle{\displaystyle\rho}}

\rho
Property / Symbols used: Q12667 / qualifier
 
xml-id: C33.S1.XMD5.m1adec
Property / Symbols used
 
Property / Symbols used: Q12668 / rank
 
Normal rank
Property / Symbols used: Q12668 / qualifier
 
Defining formula:

η 𝜂 {\displaystyle{\displaystyle\eta}}

\eta
Property / Symbols used: Q12668 / qualifier
 
xml-id: C33.S1.XMD7.m1adec
Property / Symbols used
 
Property / Symbols used: Q12687 / rank
 
Normal rank
Property / Symbols used: Q12687 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q12687 / qualifier
 
xml-id: C33.S12.XMD1.m1adec
Property / Symbols used
 
Property / Symbols used: Q12688 / rank
 
Normal rank
Property / Symbols used: Q12688 / qualifier
 
Defining formula:

μ 𝜇 {\displaystyle{\displaystyle\mu}}

\mu
Property / Symbols used: Q12688 / qualifier
 
xml-id: C33.S12.XMD2.m1adec
Property / Symbols used
 
Property / Symbols used: Q12689 / rank
 
Normal rank
Property / Symbols used: Q12689 / qualifier
 
Defining formula:

A j subscript 𝐴 𝑗 {\displaystyle{\displaystyle A_{j}}}

A_{j}
Property / Symbols used: Q12689 / qualifier
 
xml-id: C33.S12.XMD3.m1dec
Property / Symbols used
 
Property / Symbols used: Q12690 / rank
 
Normal rank
Property / Symbols used: Q12690 / qualifier
 
Defining formula:

B j subscript 𝐵 𝑗 {\displaystyle{\displaystyle B_{j}}}

B_{j}
Property / Symbols used: Q12690 / qualifier
 
xml-id: C33.S12.XMD4.m1dec

Latest revision as of 14:29, 2 January 2020

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DLMF:33.12.E2
No description defined

    Statements

    Defining formula
    F 0 ⁡ ( η , ρ ) G 0 ⁡ ( η , ρ ) ∼ π 1 / 2 ⁢ ( 2 ⁢ η ) 1 / 6 ⁢ { Ai ⁡ ( x ) Bi ⁡ ( x ) ⁢ ( 1 + B 1 μ + B 2 μ 2 + ⋯ ) + Ai ′ ⁡ ( x ) Bi ′ ⁡ ( x ) ⁢ ( A 1 μ + A 2 μ 2 + ⋯ ) } , asymptotic-expansion FRACOP regular-Coulomb-F 0 𝜂 𝜌 irregular-Coulomb-G 0 𝜂 𝜌 superscript 𝜋 1 2 superscript 2 𝜂 1 6 FRACOP Airy-Ai 𝑥 Airy-Bi 𝑥 1 subscript 𝐵 1 𝜇 subscript 𝐵 2 superscript 𝜇 2 ⋯ FRACOP diffop Airy-Ai 1 𝑥 diffop Airy-Bi 1 𝑥 subscript 𝐴 1 𝜇 subscript 𝐴 2 superscript 𝜇 2 ⋯ {\displaystyle{\displaystyle{F_{0}\left(\eta,\rho\right)\atop G_{0}\left(\eta,% \rho\right)}\sim\pi^{1/2}(2\eta)^{1/6}\left\{{\mathrm{Ai}\left(x\right)\atop% \mathrm{Bi}\left(x\right)}\left(1+\frac{B_{1}}{\mu}+\frac{B_{2}}{\mu^{2}}+% \cdots\right)+{\mathrm{Ai}'\left(x\right)\atop\mathrm{Bi}'\left(x\right)}\left% (\frac{A_{1}}{\mu}+\frac{A_{2}}{\mu^{2}}+\cdots\right)\right\},}}
    0 references
    DLMF id
    DLMF:33.12.E2
    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.m1adec
    0 references
    Q11081
    Defining formula
    Bi ⁡ ( z ) Airy-Bi 𝑧 {\displaystyle{\displaystyle\mathrm{Bi}\left(\NVar{z}\right)}}
    xml-id
    C9.S2.SS1.m2adec
    0 references
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
    Q12670
    Defining formula
    G ℓ ⁡ ( η , ρ ) irregular-Coulomb-G ℓ 𝜂 𝜌 {\displaystyle{\displaystyle G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}% \right)}}
    xml-id
    C33.S2.E11.m1adec
    0 references
    regular Coulomb radial function
    Defining formula
    F ℓ ⁡ ( η , ρ ) regular-Coulomb-F ℓ 𝜂 𝜌 {\displaystyle{\displaystyle F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}% \right)}}
    xml-id
    C33.S2.E3.m2adec
    0 references
    Q12667
    Defining formula
    ρ 𝜌 {\displaystyle{\displaystyle\rho}}
    xml-id
    C33.S1.XMD5.m1adec
    0 references
    Q12668
    Defining formula
    η 𝜂 {\displaystyle{\displaystyle\eta}}
    xml-id
    C33.S1.XMD7.m1adec
    0 references
    Q12687
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C33.S12.XMD1.m1adec
    0 references
    Q12688
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C33.S12.XMD2.m1adec
    0 references
    Q12689
    Defining formula
    A j subscript 𝐴 𝑗 {\displaystyle{\displaystyle A_{j}}}
    xml-id
    C33.S12.XMD3.m1dec
    0 references
    Q12690
    Defining formula
    B j subscript 𝐵 𝑗 {\displaystyle{\displaystyle B_{j}}}
    xml-id
    C33.S12.XMD4.m1dec
    0 references
     
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