DLMF:10.22.E12 (Q3386)

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DLMF:10.22.E12
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    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),}}
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