DLMF:10.43.E8

MaRDI QIDQ4319

• Label: DLMF:10.43.E8

Digital Library of Mathematical Functions ID 10.43.E8

${\displaystyle{\displaystyle\int_{0}^{x}e^{\pm t}t^{-\nu}I_{\nu}\left(t\right)\mathrm{d}t=-\frac{e^{\pm x}x^{-\nu+1}}{2\nu-1}(I_{\nu}\left(x\right)\mp I_{\nu-1}\left(x\right))\mp\frac{2^{-\nu+1}}{(2\nu-1)\Gamma\left(\nu\right)},}}$

Constraint(s)

${\displaystyle{\displaystyle\nu\neq\tfrac{1}{2}}}$

Symbols List

• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: gamma function
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}$: modified Bessel function of the first kind
• ${\displaystyle{\displaystyle x}}$: real variable
• ${\displaystyle{\displaystyle\nu}}$: complex parameter
 

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