DLMF:13.8.E8 (Q4433): 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: Q11566 / rank
 
Normal rank
Property / Symbols used: Q11566 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11566 / qualifier
 
xml-id: C13.S1.XMD4.m1dec

Latest revision as of 15:57, 2 January 2020

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DLMF:13.8.E8
No description defined

    Statements

    Defining formula
    U ⁡ ( a , b , x ) = 2 ⁢ e 1 2 ⁢ x Γ ⁡ ( a ) ⁢ ( 2 β ⁢ tanh ⁡ ( w 2 ) ⁢ ( 1 - e - w β ) - b ⁢ β 1 - b ⁢ K 1 - b ⁡ ( 2 ⁢ β ⁢ a ) + a - 1 ⁢ ( a - 1 + β 1 + β ) 1 - b ⁢ e - 2 ⁢ β ⁢ a ⁢ O ⁡ ( 1 ) ) , Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑥 2 superscript 𝑒 1 2 𝑥 Euler-Gamma 𝑎 2 𝛽 𝑤 2 superscript 1 superscript 𝑒 𝑤 𝛽 𝑏 superscript 𝛽 1 𝑏 modified-Bessel-second-kind 1 𝑏 2 𝛽 𝑎 superscript 𝑎 1 superscript superscript 𝑎 1 𝛽 1 𝛽 1 𝑏 superscript 𝑒 2 𝛽 𝑎 Big-O 1 {\displaystyle{\displaystyle U\left(a,b,x\right)=\frac{2e^{\frac{1}{2}x}}{% \Gamma\left(a\right)}\left(\sqrt{\frac{2}{\beta}\tanh\left(\frac{w}{2}\right)}% \left(\frac{1-e^{-w}}{\beta}\right)^{-b}\beta^{1-b}K_{1-b}\left(2\beta a\right% )+a^{-1}\left(\frac{a^{-1}+\beta}{1+\beta}\right)^{1-b}e^{-2\beta a}O\left(1% \right)\right),}}
    0 references
    DLMF id
    DLMF:13.8.E8
    0 references
    Symbols used
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2acdec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2acdec
    0 references
    Kummer confluent hypergeometric function
    Defining formula
    U ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{b},\NVar{z}\right)}}
    xml-id
    C13.S2.E6.m2abdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
    0 references
    hyperbolic tangent function
    Defining formula
    tanh ⁡ z 𝑧 {\displaystyle{\displaystyle\tanh\NVar{z}}}
    xml-id
    C4.S28.E4.m2adec
    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
    Q11566
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C13.S1.XMD4.m1dec
    0 references
     
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