DLMF:10.22.E12 (Q3386): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
test:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1akdec
Property / Symbols used
 
Property / Symbols used: psi (or digamma) function / rank
 
Normal rank
Property / Symbols used: psi (or digamma) function / qualifier
 
test:

ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}

\digamma@{\NVar{z}}
Property / Symbols used: psi (or digamma) function / qualifier
 
xml-id: C5.S2.E2.m2aadec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
test:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3akdec
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.m1ddec

Latest revision as of 14:49, 2 January 2020

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English
DLMF:10.22.E12
No description defined

    Statements

    test
    x ⁢ ∫ 0 x 1 - J 0 ⁡ ( t ) t ⁢ d t = 2 ⁢ ∑ k = 0 ∞ ( 2 ⁢ k + 3 ) ⁢ ( ψ ⁡ ( k + 2 ) - ψ ⁡ ( 1 ) ) ⁢ J 2 ⁢ k + 3 ⁡ ( x ) = x - 2 ⁢ J 1 ⁡ ( x ) + 2 ⁢ ∑ k = 0 ∞ ( 2 ⁢ k + 5 ) ⁢ ( ψ ⁡ ( k + 3 ) - ψ ⁡ ( 1 ) - 1 ) ⁢ J 2 ⁢ k + 5 ⁡ ( x ) , 𝑥 superscript subscript 0 𝑥 1 Bessel-J 0 𝑡 𝑡 𝑡 2 superscript subscript 𝑘 0 2 𝑘 3 digamma 𝑘 2 digamma 1 Bessel-J 2 𝑘 3 𝑥 𝑥 2 Bessel-J 1 𝑥 2 superscript subscript 𝑘 0 2 𝑘 5 digamma 𝑘 3 digamma 1 1 Bessel-J 2 𝑘 5 𝑥 {\displaystyle{\displaystyle x\int_{0}^{x}\frac{1-J_{0}\left(t\right)}{t}% \mathrm{d}t=2\sum_{k=0}^{\infty}(2k+3)(\psi\left(k+2\right)-\psi\left(1\right)% )J_{2k+3}\left(x\right)=x-2J_{1}\left(x\right)+2\sum_{k=0}^{\infty}(2k+5)\*(% \psi\left(k+3\right)-\psi\left(1\right)-1)J_{2k+5}\left(x\right),}}
    0 references
    DLMF id
    DLMF:10.22.E12
    0 references
    Symbols used
    Bessel function of the first kind
    test
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2addec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1akdec
    0 references
    psi (or digamma) function
    test
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2aadec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3akdec
    0 references
    Q11428
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C10.S1.XMD3.m1ddec
    0 references
     
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