DLMF:10.31.E1 (Q3518): 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
 
test:

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: principal branch of logarithm function / rank
 
Normal rank
Property / Symbols used: principal branch of logarithm function / qualifier
 
test:

ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}

\ln@@{\NVar{z}}
Property / Symbols used: principal branch of logarithm function / qualifier
 
xml-id: C4.S2.E2.m2adec
Property / Symbols used
 
Property / Symbols used: Q11429 / rank
 
Normal rank
Property / Symbols used: Q11429 / qualifier
 
test:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11429 / qualifier
 
xml-id: C10.S1.XMD2.m1dec
Property / Symbols used
 
Property / Symbols used: Q11428 / rank
 
Normal rank
Property / Symbols used: Q11428 / qualifier
 
test:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11428 / qualifier
 
xml-id: C10.S1.XMD3.m1dec
Property / Symbols used
 
Property / Symbols used: Q11426 / rank
 
Normal rank
Property / Symbols used: Q11426 / qualifier
 
test:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11426 / qualifier
 
xml-id: C10.S1.XMD6.m1dec

Latest revision as of 13:39, 2 January 2020

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DLMF:10.31.E1
No description defined

    Statements

    test
    K n ⁡ ( z ) = 1 2 ⁢ ( 1 2 ⁢ z ) - n ⁢ ∑ k = 0 n - 1 ( n - k - 1 ) ! k ! ⁢ ( - 1 4 ⁢ z 2 ) k + ( - 1 ) n + 1 ⁢ ln ⁡ ( 1 2 ⁢ z ) ⁢ I n ⁡ ( z ) + ( - 1 ) n ⁢ 1 2 ⁢ ( 1 2 ⁢ z ) n ⁢ ∑ k = 0 ∞ ( ψ ⁡ ( k + 1 ) + ψ ⁡ ( n + k + 1 ) ) ⁢ ( 1 4 ⁢ z 2 ) k k ! ⁢ ( n + k ) ! , modified-Bessel-second-kind 𝑛 𝑧 1 2 superscript 1 2 𝑧 𝑛 superscript subscript 𝑘 0 𝑛 1 𝑛 𝑘 1 𝑘 superscript 1 4 superscript 𝑧 2 𝑘 superscript 1 𝑛 1 1 2 𝑧 modified-Bessel-first-kind 𝑛 𝑧 superscript 1 𝑛 1 2 superscript 1 2 𝑧 𝑛 superscript subscript 𝑘 0 digamma 𝑘 1 digamma 𝑛 𝑘 1 superscript 1 4 superscript 𝑧 2 𝑘 𝑘 𝑛 𝑘 {\displaystyle{\displaystyle K_{n}\left(z\right)=\tfrac{1}{2}(\tfrac{1}{2}z)^{% -n}\sum_{k=0}^{n-1}\frac{(n-k-1)!}{k!}(-\tfrac{1}{4}z^{2})^{k}+(-1)^{n+1}\ln% \left(\tfrac{1}{2}z\right)I_{n}\left(z\right)+(-1)^{n}\tfrac{1}{2}(\tfrac{1}{2% }z)^{n}\sum_{k=0}^{\infty}\left(\psi\left(k+1\right)+\psi\left(n+k+1\right)% \right)\frac{(\tfrac{1}{4}z^{2})^{k}}{k!(n+k)!},}}
    0 references
    DLMF id
    DLMF:10.31.E1
    0 references
    Symbols used
    psi (or digamma) function
    test
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2adec
    0 references
    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
    0 references
    modified Bessel function of the first kind
    test
    I ν ⁡ ( z ) modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E2.m2adec
    0 references
    modified Bessel function of the second kind
    test
    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
    principal branch of logarithm function
    test
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2adec
    0 references
    Q11429
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C10.S1.XMD2.m1dec
    0 references
    Q11428
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C10.S1.XMD3.m1dec
    0 references
    Q11426
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1dec
    0 references
     
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