DLMF:10.27.E5 (Q3495): Difference between revisions

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Property / constraint
 

n = 0 , ± 1 , ± 2 , 𝑛 0 plus-or-minus 1 plus-or-minus 2 {\displaystyle{\displaystyle n=0,\pm 1,\pm 2,\ldots}}

n=0,\pm 1,\pm 2,\dotsc
Property / constraint: n = 0 , ± 1 , ± 2 , 𝑛 0 plus-or-minus 1 plus-or-minus 2 {\displaystyle{\displaystyle n=0,\pm 1,\pm 2,\ldots}} / rank
 
Normal rank
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.m2acdec
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.m2acdec
Property / Symbols used
 
Property / Symbols used: partial derivative of $$f$$ with respect to $$x$$ / rank
 
Normal rank
Property / Symbols used: partial derivative of $$f$$ with respect to $$x$$ / qualifier
 
test:

f x partial-derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\partial\NVar{f}}{\partial\NVar{x}}}}

\pderiv{\NVar{f}}{\NVar{x}}
Property / Symbols used: partial derivative of $$f$$ with respect to $$x$$ / qualifier
 
xml-id: C1.S5.E3.m4adec
Property / Symbols used
 
Property / Symbols used: partial differential of $$x$$ / rank
 
Normal rank
Property / Symbols used: partial differential of $$x$$ / qualifier
 
test:

x 𝑥 {\displaystyle{\displaystyle\partial\NVar{x}}}

\pdiff{\NVar{x}}
Property / Symbols used: partial differential of $$x$$ / qualifier
 
xml-id: C1.S5.E3.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.m1adec
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
Property / Symbols used
 
Property / Symbols used: Q11427 / rank
 
Normal rank
Property / Symbols used: Q11427 / qualifier
 
test:

ν 𝜈 {\displaystyle{\displaystyle\nu}}

\nu
Property / Symbols used: Q11427 / qualifier
 
xml-id: C10.S1.XMD7.m1cdec

Latest revision as of 12:35, 2 January 2020

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DLMF:10.27.E5
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    Statements

    K n ( z ) = ( - 1 ) n - 1 2 ( I ν ( z ) ν | ν = n + I ν ( z ) ν | ν = - n ) , modified-Bessel-second-kind 𝑛 𝑧 superscript 1 𝑛 1 2 evaluated-at partial-derivative modified-Bessel-first-kind 𝜈 𝑧 𝜈 𝜈 𝑛 evaluated-at partial-derivative modified-Bessel-first-kind 𝜈 𝑧 𝜈 𝜈 𝑛 {\displaystyle{\displaystyle K_{n}\left(z\right)=\frac{(-1)^{n-1}}{2}\*\left(% \left.\frac{\partial I_{\nu}\left(z\right)}{\partial\nu}\right|_{\nu=n}+\left.% \frac{\partial I_{\nu}\left(z\right)}{\partial\nu}\right|_{\nu=-n}\right),}}
    0 references
    0 references
    n = 0 , ± 1 , ± 2 , 𝑛 0 plus-or-minus 1 plus-or-minus 2 {\displaystyle{\displaystyle n=0,\pm 1,\pm 2,\ldots}}
    0 references
    I ν ( z ) modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}
    C10.S25.E2.m2acdec
    0 references
    K ν ( z ) modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}
    C10.S25.E3.m2acdec
    0 references
    f x partial-derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\partial\NVar{f}}{\partial\NVar{x}}}}
    C1.S5.E3.m4adec
    0 references
    x 𝑥 {\displaystyle{\displaystyle\partial\NVar{x}}}
    C1.S5.E3.m2adec
    0 references
    n 𝑛 {\displaystyle{\displaystyle n}}
    C10.S1.XMD2.m1adec
    0 references
    z 𝑧 {\displaystyle{\displaystyle z}}
    C10.S1.XMD6.m1ddec
    0 references
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    C10.S1.XMD7.m1cdec
    0 references