DLMF:5.11.E14 (Q2136)

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DLMF:5.11.E14
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    Statements

    Defining formula
    Γ ⁡ ( z + a ) Γ ⁡ ( z + b ) ∼ ( z + a + b - 1 2 ) a - b ⁢ ∑ k = 0 ∞ H k ⁢ ( a , b ) ( z + 1 2 ⁢ ( a + b - 1 ) ) 2 ⁢ k . asymptotic-expansion Euler-Gamma 𝑧 𝑎 Euler-Gamma 𝑧 𝑏 superscript 𝑧 𝑎 𝑏 1 2 𝑎 𝑏 superscript subscript 𝑘 0 subscript 𝐻 𝑘 𝑎 𝑏 superscript 𝑧 1 2 𝑎 𝑏 1 2 𝑘 {\displaystyle{\displaystyle\frac{\Gamma\left(z+a\right)}{\Gamma\left(z+b% \right)}\sim\left(z+\frac{a+b-1}{2}\right)^{a-b}\sum_{k=0}^{\infty}\frac{H_{k}% (a,b)}{\left(z+\tfrac{1}{2}(a+b-1)\right)^{2k}}.}}
    0 references
    DLMF id
    DLMF:5.11.E14
    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.m2aidec
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    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11aedec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1adec
    0 references
    Q11279
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C5.S1.XMD4.m1hdec
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    Q11273
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C5.S1.XMD7.m1idec
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    Q11275
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C5.S1.XMD8.m1bdec
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    Q11283
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C5.S1.XMD9.m1cdec
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    coefficients (locally)
    Defining formula
    H k ⁢ ( a , b ) subscript 𝐻 𝑘 𝑎 𝑏 {\displaystyle{\displaystyle H_{k}(a,b)}}
    xml-id
    C5.S11.XMD5.m1dec
    0 references
     
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