DLMF:8.11.E11 (Q2584)

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DLMF:8.11.E11
No description defined

    Statements

    Defining formula
    γ * ⁡ ( 1 - a , - x ) = x a - 1 ⁢ ( - cos ⁡ ( π ⁢ a ) + sin ⁡ ( π ⁢ a ) π ⁢ ( 2 ⁢ π ⁢ F ⁡ ( y ) + 2 3 ⁢ 2 ⁢ π a ⁢ ( 1 - y 2 ) ) ⁢ e y 2 + O ⁡ ( a - 1 ) ) , incomplete-gamma-star 1 𝑎 𝑥 superscript 𝑥 𝑎 1 𝜋 𝑎 𝜋 𝑎 𝜋 2 𝜋 Dawsons-integral 𝑦 2 3 2 𝜋 𝑎 1 superscript 𝑦 2 superscript 𝑒 superscript 𝑦 2 Big-O superscript 𝑎 1 {\displaystyle{\displaystyle\gamma^{*}\left(1-a,-x\right)=x^{a-1}\left(-\cos% \left(\pi a\right)+\frac{\sin\left(\pi a\right)}{\pi}\left(2\sqrt{\pi}F\left(y% \right)+\frac{2}{3}\sqrt{\frac{2\pi}{a}}\left(1-y^{2}\right)\right)e^{y^{2}}+O% \left(a^{-1}\right)\right),}}
    0 references
    DLMF id
    DLMF:8.11.E11
    0 references
    Symbols used
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2abdec
    0 references
    Dawson’s integral
    Defining formula
    F ⁡ ( z ) Dawsons-integral 𝑧 {\displaystyle{\displaystyle F\left(\NVar{z}\right)}}
    xml-id
    C7.S2.E5.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2afdec
    0 references
    incomplete gamma function
    Defining formula
    γ * ⁡ ( a , z ) incomplete-gamma-star 𝑎 𝑧 {\displaystyle{\displaystyle\gamma^{*}\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C8.S2.E6.m2adec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2adec
    0 references
    Q11332
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C8.S1.XMD1.m1cdec
    0 references
    Q11324
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C8.S1.XMD3.m1hdec
    0 references
    Q11346
    Defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C8.S11.XMD5.m1adec
    0 references
     
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