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 Ξ“ ⁑ ( a ) ⁒ βˆ‘ s = 0 ∞ ( 1 - a ) s ⁒ ( b - a ) s s ! ⁒ z - s + e Β± Ο€ ⁒ i ⁒ a ⁒ z - a Ξ“ ⁑ ( b - a ) ⁒ βˆ‘ s = 0 ∞ ( a ) s ⁒ ( a - b + 1 ) s s ! ⁒ ( - z ) - s , asymptotic-expansion Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑏 𝑧 superscript 𝑒 𝑧 superscript 𝑧 π‘Ž 𝑏 Euler-Gamma π‘Ž superscript subscript 𝑠 0 Pochhammer 1 π‘Ž 𝑠 Pochhammer 𝑏 π‘Ž 𝑠 𝑠 superscript 𝑧 𝑠 superscript 𝑒 plus-or-minus πœ‹ imaginary-unit π‘Ž superscript 𝑧 π‘Ž Euler-Gamma 𝑏 π‘Ž superscript subscript 𝑠 0 Pochhammer π‘Ž 𝑠 Pochhammer π‘Ž 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},}}
    0 references
    DLMF id
    DLMF:13.7.E2
    0 references
    constraint
    - 1 2 ⁒ Ο€ + Ξ΄ ≀ Β± ph ⁑ z ≀ 3 2 ⁒ Ο€ - Ξ΄ 1 2 πœ‹ 𝛿 plus-or-minus phase 𝑧 3 2 πœ‹ 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
    Ξ“ ⁑ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aadec
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    Olver’s confluent hypergeometric function
    Defining formula
    𝐌 ⁑ ( a , b , z ) Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑏 𝑧 {\displaystyle{\displaystyle{\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right% )}}
    xml-id
    C13.S2.E3.m2aadec
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer π‘Ž 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1aadec
    0 references
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    Ο€ {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
     
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