DLMF:13.23.E8 (Q4642): Difference between revisions

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‎Changed an Item: Add constraint
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Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
Defining formula:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3agdec

Revision as of 01:03, 2 January 2020

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DLMF:13.23.E8
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    Statements

    Defining formula
    1 Γ ⁡ ( 1 + 2 ⁢ μ ) ⁢ ∫ 0 ∞ cos ⁡ ( 2 ⁢ x ⁢ t ) ⁢ e - 1 2 ⁢ t 2 ⁢ t - 2 ⁢ μ - 1 ⁢ M κ , μ ⁡ ( t 2 ) ⁢ d t = π ⁢ e - 1 2 ⁢ x 2 ⁢ x μ + κ - 1 2 ⁢ Γ ⁡ ( 1 2 + μ + κ ) ⁢ W 1 2 ⁢ κ - 3 2 ⁢ μ , 1 2 ⁢ κ + 1 2 ⁢ μ ⁡ ( x 2 ) , 1 Euler-Gamma 1 2 𝜇 superscript subscript 0 2 𝑥 𝑡 superscript 𝑒 1 2 superscript 𝑡 2 superscript 𝑡 2 𝜇 1 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 superscript 𝑡 2 𝑡 𝜋 superscript 𝑒 1 2 superscript 𝑥 2 superscript 𝑥 𝜇 𝜅 1 2 Euler-Gamma 1 2 𝜇 𝜅 Whittaker-confluent-hypergeometric-W 1 2 𝜅 3 2 𝜇 1 2 𝜅 1 2 𝜇 superscript 𝑥 2 {\displaystyle{\displaystyle\frac{1}{\Gamma\left(1+2\mu\right)}\int_{0}^{% \infty}\cos\left(2xt\right)e^{-\frac{1}{2}t^{2}}t^{-2\mu-1}M_{\kappa,\mu}\left% (t^{2}\right)\mathrm{d}t=\frac{\sqrt{\pi}e^{-\frac{1}{2}x^{2}}x^{\mu+\kappa-1}% }{2\Gamma\left(\frac{1}{2}+\mu+\kappa\right)}W_{\frac{1}{2}\kappa-\frac{3}{2}% \mu,\frac{1}{2}\kappa+\frac{1}{2}\mu}\left(x^{2}\right),}}
    0 references
    DLMF id
    DLMF:13.23.E8
    0 references
    constraint
    ℜ ⁡ ( κ + μ ) > - 1 2 𝜅 𝜇 1 2 {\displaystyle{\displaystyle\Re(\kappa+\mu)>-\tfrac{1}{2}}}
    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.m2agdec
    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.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.m2acdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1agdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2agdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3agdec
    0 references
     
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