DLMF:12.14.E31 (Q4273): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11543 / rank
 
Normal rank
Property / Symbols used: Q11543 / qualifier
 
Defining formula:

W ⁡ ( a , x ) parabolic-W 𝑎 𝑥 {\displaystyle{\displaystyle W\left(\NVar{a},\NVar{x}\right)}}

\paraW@{\NVar{a}}{\NVar{x}}
Property / Symbols used: Q11543 / qualifier
 
xml-id: C12.S14.SS1.p1.m1aodec
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.m1ddec
Property / Symbols used
 
Property / Symbols used: DLMF:12.10.E23 / rank
 
Normal rank
Property / Symbols used: DLMF:12.10.E23 / qualifier
 
Defining formula:

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

\eta
Property / Symbols used: DLMF:12.10.E23 / qualifier
 
xml-id: C12.S10.XMD12.m1dec
Property / Symbols used
 
Property / Symbols used: Q11531 / rank
 
Normal rank
Property / Symbols used: Q11531 / qualifier
 
Defining formula:

𝒜 ~ s ⁢ ( t ) subscript ~ 𝒜 𝑠 𝑡 {\displaystyle{\displaystyle\widetilde{\mathcal{A}}_{s}(t)}}

\widetilde{\mathcal{A}}_{s}(t)
Property / Symbols used: Q11531 / qualifier
 
xml-id: C12.S10.XMD13.m1dec
Property / Symbols used
 
Property / Symbols used: coefficients (locally) / rank
 
Normal rank
Property / Symbols used: coefficients (locally) / qualifier
 
Defining formula:

𝒜 s ⁢ ( t ) subscript 𝒜 𝑠 𝑡 {\displaystyle{\displaystyle\mathcal{A}_{s}(t)}}

\mathcal{A}_{s}(t)
Property / Symbols used: coefficients (locally) / qualifier
 
xml-id: C12.S10.XMD2.m1bdec
Property / Symbols used
 
Property / Symbols used: DLMF:12.14.E28 / rank
 
Normal rank
Property / Symbols used: DLMF:12.14.E28 / qualifier
 
Defining formula:

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

l(\mu)
Property / Symbols used: DLMF:12.14.E28 / qualifier
 
xml-id: C12.S14.XMD13.m1ddec

Latest revision as of 15:31, 2 January 2020

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DLMF:12.14.E31
No description defined

    Statements

    Defining formula
    W ⁡ ( 1 2 ⁢ μ 2 , μ ⁢ t ⁢ 2 ) ∼ l ⁢ ( μ ) ⁢ e μ 2 ⁢ η 2 1 2 ⁢ e 1 4 ⁢ π ⁢ μ 2 ⁢ ( 1 - t 2 ) 1 4 ⁢ ∑ s = 0 ∞ ( - 1 ) s ⁢ 𝒜 ~ s ⁢ ( t ) μ 2 ⁢ s , asymptotic-expansion parabolic-W 1 2 superscript 𝜇 2 𝜇 𝑡 2 𝑙 𝜇 superscript 𝑒 superscript 𝜇 2 𝜂 superscript 2 1 2 superscript 𝑒 1 4 𝜋 superscript 𝜇 2 superscript 1 superscript 𝑡 2 1 4 superscript subscript 𝑠 0 superscript 1 𝑠 subscript ~ 𝒜 𝑠 𝑡 superscript 𝜇 2 𝑠 {\displaystyle{\displaystyle W\left(\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)% \sim\frac{l(\mu)e^{\mu^{2}\eta}}{2^{\frac{1}{2}}e^{\frac{1}{4}\pi\mu^{2}}(1-t^% {2})^{\frac{1}{4}}}\*\sum_{s=0}^{\infty}(-1)^{s}\frac{\widetilde{\cal{A}}_{s}(% t)}{\mu^{2s}},}}
    0 references
    DLMF id
    DLMF:12.14.E31
    0 references
    Symbols used
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11afdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ajdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2agdec
    0 references
    Q11543
    Defining formula
    W ⁡ ( a , x ) parabolic-W 𝑎 𝑥 {\displaystyle{\displaystyle W\left(\NVar{a},\NVar{x}\right)}}
    xml-id
    C12.S14.SS1.p1.m1aodec
    0 references
    Q11526
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C12.S1.XMD5.m1ddec
    0 references
    DLMF:12.10.E23
    Defining formula
    η 𝜂 {\displaystyle{\displaystyle\eta}}
    xml-id
    C12.S10.XMD12.m1dec
    0 references
    Q11531
    Defining formula
    𝒜 ~ s ⁢ ( t ) subscript ~ 𝒜 𝑠 𝑡 {\displaystyle{\displaystyle\widetilde{\mathcal{A}}_{s}(t)}}
    xml-id
    C12.S10.XMD13.m1dec
    0 references
    coefficients (locally)
    Defining formula
    𝒜 s ⁢ ( t ) subscript 𝒜 𝑠 𝑡 {\displaystyle{\displaystyle\mathcal{A}_{s}(t)}}
    xml-id
    C12.S10.XMD2.m1bdec
    0 references
    DLMF:12.14.E28
    Defining formula
    l ⁢ ( μ ) 𝑙 𝜇 {\displaystyle{\displaystyle l(\mu)}}
    xml-id
    C12.S14.XMD13.m1ddec
    0 references
     
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