DLMF:13.23.E6 (Q4640): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
test:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1aedec

Revision as of 00:02, 2 January 2020

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DLMF:13.23.E6
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    Statements

    test
    1 Γ ⁡ ( 1 + 2 ⁢ μ ) ⁢ 2 ⁢ π ⁢ i ⁢ ∫ - ∞ ( 0 + ) e z ⁢ t + 1 2 ⁢ t - 1 ⁢ t κ ⁢ M κ , μ ⁡ ( t - 1 ) ⁢ d t = z - κ - 1 2 Γ ⁡ ( 1 2 + μ - κ ) ⁢ I 2 ⁢ μ ⁡ ( 2 ⁢ z ) , 1 Euler-Gamma 1 2 𝜇 2 𝜋 imaginary-unit superscript subscript limit-from 0 superscript 𝑒 𝑧 𝑡 1 2 superscript 𝑡 1 superscript 𝑡 𝜅 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 superscript 𝑡 1 𝑡 superscript 𝑧 𝜅 1 2 Euler-Gamma 1 2 𝜇 𝜅 modified-Bessel-first-kind 2 𝜇 2 𝑧 {\displaystyle{\displaystyle\frac{1}{\Gamma\left(1+2\mu\right)2\pi\mathrm{i}}% \int_{-\infty}^{(0+)}e^{zt+\frac{1}{2}t^{-1}}t^{\kappa}M_{\kappa,\mu}\left(t^{% -1}\right)\mathrm{d}t=\frac{z^{-\kappa-\frac{1}{2}}}{\Gamma\left(\frac{1}{2}+% \mu-\kappa\right)}I_{2\mu}\left(2\sqrt{z}\right),}}
    0 references
    DLMF id
    DLMF:13.23.E6
    0 references
    constraint
    ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}}
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aedec
    0 references
    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.m2acdec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aedec
    0 references
     
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