DLMF:8.7.E4 (Q2530)

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DLMF:8.7.E4
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    Statements

    test
    γ ⁡ ( a , x ) = Γ ⁡ ( a ) ⁢ x 1 2 ⁢ a ⁢ e - x ⁢ ∑ n = 0 ∞ e n ⁢ ( - 1 ) ⁢ x 1 2 ⁢ n ⁢ I n + a ⁡ ( 2 ⁢ x 1 / 2 ) , incomplete-gamma 𝑎 𝑥 Euler-Gamma 𝑎 superscript 𝑥 1 2 𝑎 superscript 𝑒 𝑥 superscript subscript 𝑛 0 subscript 𝑒 𝑛 1 superscript 𝑥 1 2 𝑛 modified-Bessel-first-kind 𝑛 𝑎 2 superscript 𝑥 1 2 {\displaystyle{\displaystyle\gamma\left(a,x\right)=\Gamma\left(a\right)x^{% \frac{1}{2}a}e^{-x}\sum_{n=0}^{\infty}e_{n}(-1)x^{\frac{1}{2}n}I_{n+a}\left(% \textstyle 2x^{1/2}\right),}}
    0 references
    DLMF id
    DLMF:8.7.E4
    0 references
    constraint
    a ≠ 0 , - 1 , - 2 , … 𝑎 0 1 2 … {\displaystyle{\displaystyle a\neq 0,-1,-2,\dots}}
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    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2abdec
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    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
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    incomplete gamma function
    test
    γ ⁡ ( a , z ) incomplete-gamma 𝑎 𝑧 {\displaystyle{\displaystyle\gamma\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C8.S2.E1.m2aadec
    0 references
     
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