DLMF:13.23.E7 (Q4641): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: modified Bessel function of the second kind / rank
 
Normal rank
Property / Symbols used: modified Bessel function of the second kind / qualifier
 
Defining formula:

K ν ⁡ ( z ) modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}

\modBesselK{\NVar{\nu}}@{\NVar{z}}
Property / Symbols used: modified Bessel function of the second kind / qualifier
 
xml-id: C10.S25.E3.m2adec
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.m1afdec
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

Latest revision as of 01:03, 2 January 2020

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English
DLMF:13.23.E7
No description defined

    Statements

    Defining formula
    1 2 ⁢ π ⁢ i ⁢ ∫ - ∞ ( 0 + ) e z ⁢ t + 1 2 ⁢ t - 1 ⁢ t κ ⁢ W κ , μ ⁡ ( t - 1 ) ⁢ d t = 2 ⁢ z - κ - 1 2 Γ ⁡ ( 1 2 + μ - κ ) ⁢ Γ ⁡ ( 1 2 - μ - κ ) ⁢ K 2 ⁢ μ ⁡ ( 2 ⁢ z ) , 1 2 𝜋 imaginary-unit superscript subscript limit-from 0 superscript 𝑒 𝑧 𝑡 1 2 superscript 𝑡 1 superscript 𝑡 𝜅 Whittaker-confluent-hypergeometric-W 𝜅 𝜇 superscript 𝑡 1 𝑡 2 superscript 𝑧 𝜅 1 2 Euler-Gamma 1 2 𝜇 𝜅 Euler-Gamma 1 2 𝜇 𝜅 modified-Bessel-second-kind 2 𝜇 2 𝑧 {\displaystyle{\displaystyle\frac{1}{2\pi\mathrm{i}}\int_{-\infty}^{(0+)}e^{zt% +\frac{1}{2}t^{-1}}t^{\kappa}W_{\kappa,\mu}\left(t^{-1}\right)\mathrm{d}t=% \frac{2z^{-\kappa-\frac{1}{2}}}{\Gamma\left(\frac{1}{2}+\mu-\kappa\right)% \Gamma\left(\frac{1}{2}-\mu-\kappa\right)}K_{2\mu}\left(2\sqrt{z}\right),}}
    0 references
    DLMF id
    DLMF:13.23.E7
    0 references
    constraint
    ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>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.m2abdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1afdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2afdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3afdec
    0 references
    modified Bessel function of the second kind
    Defining formula
    K ν ⁡ ( z ) modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E3.m2adec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1afdec
    0 references
    Q11557
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1ddec
    0 references
     
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