DLMF:13.7.E2 (Q4410)

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DLMF:13.7.E2
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    Statements

    Defining formula
    𝐌 ⁡ ( a , b , z ) ∼ e z ⁢ z a - b Γ ⁡ ( a ) ⁢ ∑ s = 0 ∞ ( 1 - a ) s ⁢ ( b - a ) s s ! ⁢ z - s + e ± π ⁢ i ⁢ a ⁢ z - a Γ ⁡ ( b - a ) ⁢ ∑ s = 0 ∞ ( a ) s ⁢ ( a - b + 1 ) s s ! ⁢ ( - z ) - s , asymptotic-expansion Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 𝑧 superscript 𝑒 𝑧 superscript 𝑧 𝑎 𝑏 Euler-Gamma 𝑎 superscript subscript 𝑠 0 Pochhammer 1 𝑎 𝑠 Pochhammer 𝑏 𝑎 𝑠 𝑠 superscript 𝑧 𝑠 superscript 𝑒 plus-or-minus 𝜋 imaginary-unit 𝑎 superscript 𝑧 𝑎 Euler-Gamma 𝑏 𝑎 superscript subscript 𝑠 0 Pochhammer 𝑎 𝑠 Pochhammer 𝑎 𝑏 1 𝑠 𝑠 superscript 𝑧 𝑠 {\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)\sim\frac{e^{z}z^{a-% b}}{\Gamma\left(a\right)}\sum_{s=0}^{\infty}\frac{{\left(1-a\right)_{s}}{\left% (b-a\right)_{s}}}{s!}z^{-s}+\frac{e^{\pm\pi\mathrm{i}a}z^{-a}}{\Gamma\left(b-a% \right)}\sum_{s=0}^{\infty}\frac{{\left(a\right)_{s}}{\left(a-b+1\right)_{s}}}% {s!}(-z)^{-s},}}
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    DLMF id
    DLMF:13.7.E2
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    constraint
    - 1 2 ⁢ π + δ ≤ ± ph ⁡ z ≤ 3 2 ⁢ π - δ 1 2 𝜋 𝛿 plus-or-minus phase 𝑧 3 2 𝜋 𝛿 {\displaystyle{\displaystyle-\frac{1}{2}\pi+\delta\leq\pm\operatorname{ph}z% \leq\frac{3}{2}\pi-\delta}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aadec
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