DLMF:12.10.E19 (Q4181): Difference between revisions

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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.m1fdec

Latest revision as of 15:19, 2 January 2020

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DLMF:12.10.E19
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    Statements

    Defining formula
    U ′ ⁡ ( - 1 2 ⁢ μ 2 , μ ⁢ t ⁢ 2 ) ∼ μ ⁢ 2 ⁢ g ⁢ ( μ ) ⁢ ( 1 - t 2 ) 1 4 ⁢ ( sin ⁡ κ ⁢ ∑ s = 0 ∞ ( - 1 ) s ⁢ ℬ ~ 2 ⁢ s ⁢ ( t ) μ 4 ⁢ s + cos ⁡ κ ⁢ ∑ s = 0 ∞ ( - 1 ) s ⁢ ℬ ~ 2 ⁢ s + 1 ⁢ ( t ) μ 4 ⁢ s + 2 ) , asymptotic-expansion diffop parabolic-U 1 1 2 superscript 𝜇 2 𝜇 𝑡 2 𝜇 2 𝑔 𝜇 superscript 1 superscript 𝑡 2 1 4 𝜅 superscript subscript 𝑠 0 superscript 1 𝑠 subscript ~ ℬ 2 𝑠 𝑡 superscript 𝜇 4 𝑠 𝜅 superscript subscript 𝑠 0 superscript 1 𝑠 subscript ~ ℬ 2 𝑠 1 𝑡 superscript 𝜇 4 𝑠 2 {\displaystyle{\displaystyle U'\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)% \sim\mu\sqrt{2}g(\mu)(1-t^{2})^{\frac{1}{4}}\*\left(\sin\kappa\sum_{s=0}^{% \infty}(-1)^{s}\frac{\widetilde{\cal{B}}_{2s}(t)}{\mu^{4s}}+\cos\kappa\sum_{s=% 0}^{\infty}(-1)^{s}\frac{\widetilde{\cal{B}}_{2s+1}(t)}{\mu^{4s+2}}\right),}}
    0 references
    DLMF id
    DLMF:12.10.E19
    0 references
    Symbols used
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11agdec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aadec
    0 references
    Q11320
    Defining formula
    U ⁡ ( a , z ) parabolic-U 𝑎 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C12.S2.SS1.p1.m1acdec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aadec
    0 references
    Q11526
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C12.S1.XMD5.m1mdec
    0 references
    Q11530
    Defining formula
    κ 𝜅 {\displaystyle{\displaystyle\kappa}}
    xml-id
    C12.S10.XMD10.m1adec
    0 references
    Q11532
    Defining formula
    ℬ ~ s ⁢ ( t ) subscript ~ ℬ 𝑠 𝑡 {\displaystyle{\displaystyle\widetilde{\mathcal{B}}_{s}(t)}}
    xml-id
    C12.S10.XMD14.m1dec
    0 references
    expansion (locally)
    Defining formula
    g ⁢ ( μ ) 𝑔 𝜇 {\displaystyle{\displaystyle g(\mu)}}
    xml-id
    C12.S10.XMD7.m1fdec
    0 references
     
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