DLMF:13.19.E1 (Q4587)

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DLMF:13.19.E1
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    Statements

    Defining formula
    M κ , μ ⁡ ( x ) ∼ Γ ⁡ ( 1 + 2 ⁢ μ ) Γ ⁡ ( 1 2 + μ - κ ) ⁢ e 1 2 ⁢ x ⁢ x - κ ⁢ ∑ s = 0 ∞ ( 1 2 - μ + κ ) s ⁢ ( 1 2 + μ + κ ) s s ! ⁢ x - s , asymptotic-expansion Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑥 Euler-Gamma 1 2 𝜇 Euler-Gamma 1 2 𝜇 𝜅 superscript 𝑒 1 2 𝑥 superscript 𝑥 𝜅 superscript subscript 𝑠 0 Pochhammer 1 2 𝜇 𝜅 𝑠 Pochhammer 1 2 𝜇 𝜅 𝑠 𝑠 superscript 𝑥 𝑠 {\displaystyle{\displaystyle M_{\kappa,\mu}\left(x\right)\sim\frac{\Gamma\left% (1+2\mu\right)}{\Gamma\left(\frac{1}{2}+\mu-\kappa\right)}e^{\frac{1}{2}x}x^{-% \kappa}\*\sum_{s=0}^{\infty}\frac{{\left(\frac{1}{2}-\mu+\kappa\right)_{s}}{% \left(\frac{1}{2}+\mu+\kappa\right)_{s}}}{s!}x^{-s},}}
    0 references
    DLMF id
    DLMF:13.19.E1
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    constraint
    μ - κ ≠ - 1 2 , - 3 2 , … 𝜇 𝜅 1 2 3 2 … {\displaystyle{\displaystyle\mu-\kappa\neq-\tfrac{1}{2},-\tfrac{3}{2},\dots}}
    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.m2adec
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    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1adec
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    Whittaker confluent hypergeometric function
    Defining formula
    M κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}
    xml-id
    C13.S14.E2.m2adec
    0 references
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11adec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
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    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
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    Q11558
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C13.S1.XMD3.m1dec
    0 references
     
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