DLMF:8.18.E14 (Q2682)

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DLMF:8.18.E14
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    Statements

    Defining formula
    I x ⁡ ( a , b ) ∼ Q ⁡ ( b , a ⁢ ζ ) - ( 2 ⁢ π ⁢ b ) - 1 / 2 Γ ~ ⁢ ( b ) ⁢ ( x x 0 ) a ⁢ ( 1 - x 1 - x 0 ) b ⁢ ∑ k = 0 ∞ h k ⁢ ( ζ , μ ) a k , asymptotic-expansion IncI 𝑥 𝑎 𝑏 incomplete-gamma-Q 𝑏 𝑎 𝜁 superscript 2 𝜋 𝑏 1 2 ~ Γ 𝑏 superscript 𝑥 subscript 𝑥 0 𝑎 superscript 1 𝑥 1 subscript 𝑥 0 𝑏 superscript subscript 𝑘 0 subscript ℎ 𝑘 𝜁 𝜇 superscript 𝑎 𝑘 {\displaystyle{\displaystyle I_{x}\left(a,b\right)\sim Q\left(b,a\zeta\right)-% \frac{(2\pi b)^{-1/2}}{\widetilde{\Gamma}(b)}\left(\frac{x}{x_{0}}\right)^{a}% \left(\frac{1-x}{1-x_{0}}\right)^{b}\sum_{k=0}^{\infty}\frac{h_{k}(\zeta,\mu)}% {a^{k}},}}
    0 references
    DLMF id
    DLMF:8.18.E14
    0 references
    Symbols used
    incomplete beta function
    Defining formula
    I x ⁡ ( a , b ) IncI 𝑥 𝑎 𝑏 {\displaystyle{\displaystyle I_{\NVar{x}}\left(\NVar{a},\NVar{b}\right)}}
    xml-id
    C8.S17.E2.m2acdec
    0 references
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11abdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    Q11326
    Defining formula
    Q ⁡ ( a , z ) incomplete-gamma-Q 𝑎 𝑧 {\displaystyle{\displaystyle Q\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C8.S2.E4.m2aadec
    0 references
     
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