DLMF:13.23.E8 (Q4642)

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DLMF:13.23.E8
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    Statements

    test
    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),}}
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    DLMF id
    DLMF:13.23.E8
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    constraint
    ℜ ⁡ ( κ + μ ) > - 1 2 𝜅 𝜇 1 2 {\displaystyle{\displaystyle\Re(\kappa+\mu)>-\tfrac{1}{2}}}
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    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2agdec
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    Whittaker confluent hypergeometric function
    test
    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
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    Whittaker confluent hypergeometric function
    test
    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
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    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
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