DLMF:10.44.E6 (Q3654): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: modified Bessel function of the first kind / rank
 
Normal rank
Property / Symbols used: modified Bessel function of the first kind / qualifier
 
test:

I ν ⁡ ( z ) modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}

\modBesselI{\NVar{\nu}}@{\NVar{z}}
Property / Symbols used: modified Bessel function of the first kind / qualifier
 
xml-id: C10.S25.E2.m2addec
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.m2aadec
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.m2aadec
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.m1edec
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.m1ddec

Latest revision as of 13:05, 2 January 2020

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DLMF:10.44.E6
No description defined

    Statements

    test
    K n ⁡ ( z ) = n ! ⁢ ( 1 2 ⁢ z ) - n 2 ⁢ ∑ k = 0 n - 1 ( - 1 ) k ⁢ ( 1 2 ⁢ z ) k ⁢ I k ⁡ ( z ) k ! ⁢ ( n - k ) + ( - 1 ) n - 1 ⁢ ( ln ⁡ ( 1 2 ⁢ z ) - ψ ⁡ ( n + 1 ) ) ⁢ I n ⁡ ( z ) + ( - 1 ) n ⁢ ∑ k = 1 ∞ ( n + 2 ⁢ k ) ⁢ I n + 2 ⁢ k ⁡ ( z ) k ⁢ ( n + k ) , modified-Bessel-second-kind 𝑛 𝑧 𝑛 superscript 1 2 𝑧 𝑛 2 superscript subscript 𝑘 0 𝑛 1 superscript 1 𝑘 superscript 1 2 𝑧 𝑘 modified-Bessel-first-kind 𝑘 𝑧 𝑘 𝑛 𝑘 superscript 1 𝑛 1 1 2 𝑧 digamma 𝑛 1 modified-Bessel-first-kind 𝑛 𝑧 superscript 1 𝑛 superscript subscript 𝑘 1 𝑛 2 𝑘 modified-Bessel-first-kind 𝑛 2 𝑘 𝑧 𝑘 𝑛 𝑘 {\displaystyle{\displaystyle K_{n}\left(z\right)=\frac{n!(\tfrac{1}{2}z)^{-n}}% {2}\sum_{k=0}^{n-1}(-1)^{k}\frac{(\tfrac{1}{2}z)^{k}I_{k}\left(z\right)}{k!(n-% k)}+(-1)^{n-1}\left(\ln\left(\tfrac{1}{2}z\right)-\psi\left(n+1\right)\right)I% _{n}\left(z\right)+(-1)^{n}\sum_{k=1}^{\infty}\frac{(n+2k)I_{n+2k}\left(z% \right)}{k(n+k)},}}
    0 references
    DLMF id
    DLMF:10.44.E6
    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.m5acdec
    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.m2addec
    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.m2aadec
    0 references
    principal branch of logarithm function
    test
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2aadec
    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.m1edec
    0 references
    Q11426
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1ddec
    0 references
     
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