DLMF:13.14.E18 (Q4507): Difference between revisions

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Property / constraint
 

μ ≠ 0 𝜇 0 {\displaystyle{\displaystyle\mu\not=0}}

\mu\not=0
Property / constraint: μ ≠ 0 𝜇 0 {\displaystyle{\displaystyle\mu\not=0}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: gamma function / rank
 
Normal rank
Property / Symbols used: gamma function / qualifier
 
Defining formula:

Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}

\EulerGamma@{\NVar{z}}
Property / Symbols used: gamma function / qualifier
 
xml-id: C5.S2.E1.m2afdec
Property / Symbols used
 
Property / Symbols used: Whittaker confluent hypergeometric function / rank
 
Normal rank
Property / Symbols used: Whittaker confluent hypergeometric function / qualifier
 
Defining formula:

W κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}

\WhittakerconfhyperW{\NVar{\kappa}}{\NVar{\mu}}@{\NVar{z}}
Property / Symbols used: Whittaker confluent hypergeometric function / qualifier
 
xml-id: C13.S14.E3.m2aidec
Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
Defining formula:

ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1aadec
Property / Symbols used
 
Property / Symbols used: Q11557 / rank
 
Normal rank
Property / Symbols used: Q11557 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11557 / qualifier
 
xml-id: C13.S1.XMD6.m1qdec
Property / Symbols used
 
Property / Symbols used: order not exceeding / rank
 
Normal rank
Property / Symbols used: order not exceeding / qualifier
 
Defining formula:

O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}

\bigO@{\NVar{x}}
Property / Symbols used: order not exceeding / qualifier
 
xml-id: C2.S1.E3.m2addec

Latest revision as of 16:06, 2 January 2020

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DLMF:13.14.E18
No description defined

    Statements

    Defining formula
    W κ , μ ⁡ ( z ) = Γ ⁡ ( 2 ⁢ μ ) Γ ⁡ ( 1 2 + μ - κ ) ⁢ z 1 2 - μ + Γ ⁡ ( - 2 ⁢ μ ) Γ ⁡ ( 1 2 - μ - κ ) ⁢ z 1 2 + μ + O ⁡ ( z 3 2 - ℜ ⁡ μ ) , Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 Euler-Gamma 2 𝜇 Euler-Gamma 1 2 𝜇 𝜅 superscript 𝑧 1 2 𝜇 Euler-Gamma 2 𝜇 Euler-Gamma 1 2 𝜇 𝜅 superscript 𝑧 1 2 𝜇 Big-O superscript 𝑧 3 2 𝜇 {\displaystyle{\displaystyle W_{\kappa,\mu}\left(z\right)=\frac{\Gamma\left(2% \mu\right)}{\Gamma\left(\frac{1}{2}+\mu-\kappa\right)}z^{\frac{1}{2}-\mu}+% \frac{\Gamma\left(-2\mu\right)}{\Gamma\left(\frac{1}{2}-\mu-\kappa\right)}z^{% \frac{1}{2}+\mu}+O\left(z^{\frac{3}{2}-\Re\mu}\right),}}
    0 references
    DLMF id
    DLMF:13.14.E18
    0 references
    constraint
    0 ≤ ℜ ⁡ μ < 1 2 0 𝜇 1 2 {\displaystyle{\displaystyle 0\leq\Re\mu<\tfrac{1}{2}}}
    0 references
    0 ≤ ℜ ⁡ μ < 1 2 0 𝜇 1 2 {\displaystyle{\displaystyle 0\leq\Re\mu<\tfrac{1}{2}}}
    0 references
    μ ≠ 0 𝜇 0 {\displaystyle{\displaystyle\mu\not=0}}
    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.m2afdec
    0 references
    Whittaker confluent hypergeometric function
    Defining formula
    W κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}
    xml-id
    C13.S14.E3.m2aidec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aadec
    0 references
    Q11557
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1qdec
    0 references
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2addec
    0 references
     
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