DLMF:13.16.E5 (Q4556): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
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.m1ddec

Revision as of 16:10, 2 January 2020

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DLMF:13.16.E5
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    Statements

    Defining formula
    W κ , μ ⁡ ( z ) = z μ + 1 2 ⁢ 2 - 2 ⁢ μ Γ ⁡ ( 1 2 + μ - κ ) ⁢ ∫ 1 ∞ e - 1 2 ⁢ z ⁢ t ⁢ ( t - 1 ) μ - 1 2 - κ ⁢ ( t + 1 ) μ - 1 2 + κ ⁢ d t , Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 superscript 𝑧 𝜇 1 2 superscript 2 2 𝜇 Euler-Gamma 1 2 𝜇 𝜅 superscript subscript 1 superscript 𝑒 1 2 𝑧 𝑡 superscript 𝑡 1 𝜇 1 2 𝜅 superscript 𝑡 1 𝜇 1 2 𝜅 𝑡 {\displaystyle{\displaystyle W_{\kappa,\mu}\left(z\right)=\frac{z^{\mu+\frac{1% }{2}}2^{-2\mu}}{\Gamma\left(\frac{1}{2}+\mu-\kappa\right)}\*\int_{1}^{\infty}e% ^{-\frac{1}{2}zt}(t-1)^{\mu-\frac{1}{2}-\kappa}(t+1)^{\mu-\frac{1}{2}+\kappa}% \mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:13.16.E5
    0 references
    constraint
    ℜ ⁡ μ + 1 2 > ℜ ⁡ κ 𝜇 1 2 𝜅 {\displaystyle{\displaystyle\Re\mu+\tfrac{1}{2}>\Re\kappa}}
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    | ph ⁡ z | < 1 2 ⁢ π phase 𝑧 1 2 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}{z}|<\frac{1}{2}\pi}}
    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.m2addec
    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.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1addec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2addec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3addec
    0 references
    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1adec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1addec
    0 references
    Q11557
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1ddec
    0 references
     
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