DLMF:10.43.E4 (Q3618)

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DLMF:10.43.E4
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    0 x I 0 ( t ) - 1 t d t = 1 2 k = 1 ( - 1 ) k - 1 ψ ( k + 1 ) - ψ ( 1 ) k ! ( 1 2 x ) k I k ( x ) = 2 x k = 0 ( - 1 ) k ( 2 k + 3 ) ( ψ ( k + 2 ) - ψ ( 1 ) ) I 2 k + 3 ( x ) . superscript subscript 0 𝑥 modified-Bessel-first-kind 0 𝑡 1 𝑡 𝑡 1 2 superscript subscript 𝑘 1 superscript 1 𝑘 1 digamma 𝑘 1 digamma 1 𝑘 superscript 1 2 𝑥 𝑘 modified-Bessel-first-kind 𝑘 𝑥 2 𝑥 superscript subscript 𝑘 0 superscript 1 𝑘 2 𝑘 3 digamma 𝑘 2 digamma 1 modified-Bessel-first-kind 2 𝑘 3 𝑥 {\displaystyle{\displaystyle\int_{0}^{x}\frac{I_{0}\left(t\right)-1}{t}\mathrm% {d}t=\frac{1}{2}\sum_{k=1}^{\infty}(-1)^{k-1}\frac{\psi\left(k+1\right)-\psi% \left(1\right)}{k!}(\tfrac{1}{2}x)^{k}I_{k}\left(x\right)=\frac{2}{x}\sum_{k=0% }^{\infty}(-1)^{k}(2k+3)(\psi\left(k+2\right)-\psi\left(1\right))I_{2k+3}\left% (x\right).}}
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    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    C1.S4.SS4.m1acdec
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    ψ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    C5.S2.E2.m2adec
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    ! {\displaystyle{\displaystyle!}}
    introduction.Sx4.p1.t1.r15.m5adec
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    {\displaystyle{\displaystyle\int}}
    C1.S4.SS4.m3acdec
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    I ν ( z ) modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}
    C10.S25.E2.m2adec
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