DLMF:12.10.E20 (Q4182)

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DLMF:12.10.E20
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    Statements

    Defining formula
    V ⁡ ( - 1 2 ⁢ μ 2 , μ ⁢ t ⁢ 2 ) ∼ 2 ⁢ g ⁢ ( μ ) Γ ⁡ ( 1 2 + 1 2 ⁢ μ 2 ) ⁢ ( 1 - t 2 ) 1 4 ⁢ ( cos ⁡ χ ⁢ ∑ s = 0 ∞ ( - 1 ) s ⁢ 𝒜 ~ 2 ⁢ s ⁢ ( t ) μ 4 ⁢ s - sin ⁡ χ ⁢ ∑ s = 0 ∞ ( - 1 ) s ⁢ 𝒜 ~ 2 ⁢ s + 1 ⁢ ( t ) μ 4 ⁢ s + 2 ) , asymptotic-expansion parabolic-V 1 2 superscript 𝜇 2 𝜇 𝑡 2 2 𝑔 𝜇 Euler-Gamma 1 2 1 2 superscript 𝜇 2 superscript 1 superscript 𝑡 2 1 4 𝜒 superscript subscript 𝑠 0 superscript 1 𝑠 subscript ~ 𝒜 2 𝑠 𝑡 superscript 𝜇 4 𝑠 𝜒 superscript subscript 𝑠 0 superscript 1 𝑠 subscript ~ 𝒜 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{2g(\mu)}{\Gamma\left(\tfrac{1}{2}+\tfrac{1}{2}\mu^{2}\right)(1-t^{2}% )^{\frac{1}{4}}}\*\left(\cos\chi\sum_{s=0}^{\infty}(-1)^{s}\frac{\widetilde{% \cal{A}}_{2s}(t)}{\mu^{4s}}-\sin\chi\sum_{s=0}^{\infty}(-1)^{s}\frac{% \widetilde{\cal{A}}_{2s+1}(t)}{\mu^{4s+2}}\right),}}
    0 references
    DLMF id
    DLMF:12.10.E20
    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.m2acdec
    0 references
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11ahdec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2abdec
    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.m2abdec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2abdec
    0 references
    Q11526
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C12.S1.XMD5.m1ndec
    0 references
    Q11533
    Defining formula
    χ 𝜒 {\displaystyle{\displaystyle\chi}}
    xml-id
    C12.S10.XMD11.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.m1adec
    0 references
    expansion (locally)
    Defining formula
    g ⁢ ( μ ) 𝑔 𝜇 {\displaystyle{\displaystyle g(\mu)}}
    xml-id
    C12.S10.XMD7.m1gdec
    0 references
     
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