DLMF:13.14.E7 (Q4496): Difference between revisions

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‎Changed an Item: Add constraint
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Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
Defining formula:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2aedec

Revision as of 16:05, 2 January 2020

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DLMF:13.14.E7
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    Statements

    Defining formula
    lim 2 ⁢ μ → - n - 1 ⁡ M κ , μ ⁡ ( z ) Γ ⁡ ( 2 ⁢ μ + 1 ) = ( - 1 2 ⁢ n - κ ) n + 1 ( n + 1 ) ! ⁢ M κ , 1 2 ⁢ ( n + 1 ) ⁡ ( z ) = e - 1 2 ⁢ z ⁢ z - 1 2 ⁢ n ⁢ ∑ s = n + 1 ∞ ( - 1 2 ⁢ n - κ ) s Γ ⁡ ( s - n ) ⁢ s ! ⁢ z s . subscript → 2 𝜇 𝑛 1 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 Euler-Gamma 2 𝜇 1 Pochhammer 1 2 𝑛 𝜅 𝑛 1 𝑛 1 Whittaker-confluent-hypergeometric-M 𝜅 1 2 𝑛 1 𝑧 superscript 𝑒 1 2 𝑧 superscript 𝑧 1 2 𝑛 superscript subscript 𝑠 𝑛 1 Pochhammer 1 2 𝑛 𝜅 𝑠 Euler-Gamma 𝑠 𝑛 𝑠 superscript 𝑧 𝑠 {\displaystyle{\displaystyle\lim_{2\mu\to-n-1}\frac{M_{\kappa,\mu}\left(z% \right)}{\Gamma\left(2\mu+1\right)}=\frac{{\left(-\frac{1}{2}n-\kappa\right)_{% n+1}}}{(n+1)!}M_{\kappa,\frac{1}{2}(n+1)}\left(z\right)=e^{-\frac{1}{2}z}z^{-% \frac{1}{2}n}\sum_{s=n+1}^{\infty}\frac{{\left(-\frac{1}{2}n-\kappa\right)_{s}% }}{\Gamma\left(s-n\right)s!}z^{s}.}}
    0 references
    DLMF id
    DLMF:13.14.E7
    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
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1aadec
    0 references
    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.m2acdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aedec
    0 references
     
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