DLMF:10.43.E5 (Q3619): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
test:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2aadec

Revision as of 13:59, 2 January 2020

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DLMF:10.43.E5
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    Statements

    test
    ∫ x ∞ K 0 ⁡ ( t ) t ⁢ d t = 1 2 ⁢ ( ln ⁡ ( 1 2 ⁢ x ) + γ ) 2 + π 2 24 - ∑ k = 1 ∞ ( ψ ⁡ ( k + 1 ) + 1 2 ⁢ k - ln ⁡ ( 1 2 ⁢ x ) ) ⁢ ( 1 2 ⁢ x ) 2 ⁢ k 2 ⁢ k ⁢ ( k ! ) 2 , superscript subscript 𝑥 modified-Bessel-second-kind 0 𝑡 𝑡 𝑡 1 2 superscript 1 2 𝑥 2 superscript 𝜋 2 24 superscript subscript 𝑘 1 digamma 𝑘 1 1 2 𝑘 1 2 𝑥 superscript 1 2 𝑥 2 𝑘 2 𝑘 superscript 𝑘 2 {\displaystyle{\displaystyle\int_{x}^{\infty}\frac{K_{0}\left(t\right)}{t}% \mathrm{d}t=\frac{1}{2}\left(\ln\left(\tfrac{1}{2}x\right)+\gamma\right)^{2}+% \frac{\pi^{2}}{24}-\sum_{k=1}^{\infty}\left(\psi\left(k+1\right)+\frac{1}{2k}-% \ln\left(\tfrac{1}{2}x\right)\right)\frac{(\tfrac{1}{2}x)^{2k}}{2k(k!)^{2}},}}
    0 references
    DLMF id
    DLMF:10.43.E5
    0 references
    Symbols used
    Euler’s constant
    test
    γ {\displaystyle{\displaystyle\gamma}}
    xml-id
    C5.S2.E3.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
     
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