DLMF:9.12.E15 (Q2943)

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DLMF:9.12.E15
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    Statements

    Defining formula
    Gi ⁡ ( z ) = 3 - 2 / 3 π ⁢ ∑ k = 0 ∞ cos ⁡ ( 2 ⁢ k - 1 3 ⁢ π ) ⁢ Γ ⁡ ( k + 1 3 ) ⁢ ( 3 1 / 3 ⁢ z ) k k ! , Scorer-Gi 𝑧 superscript 3 2 3 𝜋 superscript subscript 𝑘 0 2 𝑘 1 3 𝜋 Euler-Gamma 𝑘 1 3 superscript superscript 3 1 3 𝑧 𝑘 𝑘 {\displaystyle{\displaystyle\mathrm{Gi}\left(z\right)=\frac{3^{-2/3}}{\pi}\*% \sum_{k=0}^{\infty}\cos\left(\frac{2k-1}{3}\pi\right)\Gamma\left(\frac{k+1}{3}% \right)\frac{(3^{1/3}z)^{k}}{k!},}}
    0 references
    DLMF id
    DLMF:9.12.E15
    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.m2abdec
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    Scorer function (inhomogeneous Airy function)
    Defining formula
    Gi ⁡ ( z ) Scorer-Gi 𝑧 {\displaystyle{\displaystyle\mathrm{Gi}\left(\NVar{z}\right)}}
    xml-id
    C9.S12.E4.m2aidec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ahdec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
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    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
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    Q11385
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C9.S1.XMD1.m1dec
    0 references
    Q11381
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C9.S1.XMD3.m1ldec
    0 references
     
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