DLMF:12.10.E21 (Q4183): 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.m1odec
Property / Symbols used
 
Property / Symbols used: Q11533 / rank
 
Normal rank
Property / Symbols used: Q11533 / qualifier
 
Defining formula:

χ 𝜒 {\displaystyle{\displaystyle\chi}}

\chi
Property / Symbols used: Q11533 / qualifier
 
xml-id: C12.S10.XMD11.m1adec
Property / Symbols used
 
Property / Symbols used: Q11532 / rank
 
Normal rank
Property / Symbols used: Q11532 / qualifier
 
Defining formula:

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

\widetilde{\mathcal{B}}_{s}(t)
Property / Symbols used: Q11532 / qualifier
 
xml-id: C12.S10.XMD14.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.m1hdec

Latest revision as of 15:19, 2 January 2020

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DLMF:12.10.E21
No description defined

    Statements

    Defining formula
    V ′ ⁡ ( - 1 2 ⁢ μ 2 , μ ⁢ t ⁢ 2 ) ∼ μ ⁢ 2 ⁢ g ⁢ ( μ ) ⁢ ( 1 - t 2 ) 1 4 Γ ⁡ ( 1 2 + 1 2 ⁢ μ 2 ) ⁢ ( sin ⁡ χ ⁢ ∑ s = 0 ∞ ( - 1 ) s ⁢ ℬ ~ 2 ⁢ s ⁢ ( t ) μ 4 ⁢ s + cos ⁡ χ ⁢ ∑ s = 0 ∞ ( - 1 ) s ⁢ ℬ ~ 2 ⁢ s + 1 ⁢ ( t ) μ 4 ⁢ s + 2 ) , asymptotic-expansion diffop parabolic-V 1 1 2 superscript 𝜇 2 𝜇 𝑡 2 𝜇 2 𝑔 𝜇 superscript 1 superscript 𝑡 2 1 4 Euler-Gamma 1 2 1 2 superscript 𝜇 2 𝜒 superscript subscript 𝑠 0 superscript 1 𝑠 subscript ~ ℬ 2 𝑠 𝑡 superscript 𝜇 4 𝑠 𝜒 superscript subscript 𝑠 0 superscript 1 𝑠 subscript ~ ℬ 2 𝑠 1 𝑡 superscript 𝜇 4 𝑠 2 {\displaystyle{\displaystyle V'\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)% \sim\frac{\mu\sqrt{2}g(\mu)(1-t^{2})^{\frac{1}{4}}}{\Gamma\left(\tfrac{1}{2}+% \tfrac{1}{2}\mu^{2}\right)}\*\left(\sin\chi\sum_{s=0}^{\infty}(-1)^{s}\frac{% \widetilde{\cal{B}}_{2s}(t)}{\mu^{4s}}+\cos\chi\sum_{s=0}^{\infty}(-1)^{s}% \frac{\widetilde{\cal{B}}_{2s+1}(t)}{\mu^{4s+2}}\right),}}
    0 references
    DLMF id
    DLMF:12.10.E21
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2addec
    0 references
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11aidec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2acdec
    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.m2acdec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2acdec
    0 references
    Q11526
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C12.S1.XMD5.m1odec
    0 references
    Q11533
    Defining formula
    χ 𝜒 {\displaystyle{\displaystyle\chi}}
    xml-id
    C12.S10.XMD11.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.m1adec
    0 references
    expansion (locally)
    Defining formula
    g ⁢ ( μ ) 𝑔 𝜇 {\displaystyle{\displaystyle g(\mu)}}
    xml-id
    C12.S10.XMD7.m1hdec
    0 references
     
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