Formula:4315

Digital Library of Mathematical Functions ID 10.43.E4

${\displaystyle{\displaystyle\int_{0}^{x}\frac{I_{0}\left(t\right)-1}{t}\mathrm{d}t=\frac{1}{2}\sum_{k=1}^{\infty}(-1)^{k-1}\frac{\psi\left(k+1\right)-\psi\left(1\right)}{k!}(\tfrac{1}{2}x)^{k}I_{k}\left(x\right)=\frac{2}{x}\sum_{k=0}^{\infty}(-1)^{k}(2k+3)(\psi\left(k+2\right)-\psi\left(1\right))I_{2k+3}\left(x\right).}}$

Symbols List

• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}$: psi (or digamma) function
• ${\displaystyle{\displaystyle!}}$:
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}$: modified Bessel function of the first kind
• ${\displaystyle{\displaystyle k}}$: nonnegative integer
• ${\displaystyle{\displaystyle x}}$: real variable
 

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