On certain definite integrals involving the exponential-integral. (Q1547727)

Das Exponentialintegral wird definirt für positive und negative \(x\) durch \[ \text{Ei} x = \gamma + \frac {1}{4} \log{} (x^{4}) + \sum _{k = 1}^{k = \infty} \frac {x^{k}}{k . k !}, \text{wo} \; \gamma = - \varGamma (1). \] Diese Function wird nach Multiplication mit Exponentialfunctionen integrirt, und folgende Formeln werden gefunden: \[ \int _{0}^{\infty} \text{Ei} (- a x^{n} = - \varGamma \left ( \frac {1}{n} \right ) a^{- \frac {1}{n}} \] \[ \int _{0}^{\infty} e^{- a x} \text{Si} b x d x = \frac {1}{a} \text{arctg} \frac {b}{a} \] \[ \int _{0}^{\infty} e^{- a x} \text{Ci} b x d x = - \frac {1}{2 a} \log{} \left ( 1 + \frac {a^{2}}{b^{2}} \right ) \] \[ \int _{0}^{\infty} \text{Ei} (- a x) e^{- b x} d x = - \frac {1}{b} \left ( 1 + \frac {b}{a} \right ) \] \[ \int _{0}^{\infty} \text{Ei} (- a x) \sin{} b x dx = - \frac {1}{2 b} \log{} \left ( 1 + \frac {b^{2}}{a^{2}} \right ) \] \[ \int _{0}^{\infty} \text{Ei} (- a x) \cos{} b x d x = -\frac {1}{b} \text{arctg} \frac {b}{a} \] \[ \int _{0}^{\infty} \text{Ei} (- a x) \text{Si} b x d x = - \frac {1}{a} \text{arctg} \frac {b}{a} + \frac {1}{2 b} \log{} \left ( 1 + \frac {b^{2}}{a^{2}} \right ) \] \[ \int _{0}^{\infty} \text{Ei} (- a x) \text{Ci} b x d x = \frac {1}{b} \text{arctg} \frac {b}{a} + \frac {1}{2 a} \log{} \left ( 1 + \frac {a^{2}}{b^{2}} \right ) \] \[ \int _{0}^{\infty} \text{Ei} (- a x) \text{Ei} (- b x) d x = \log{} \left \{ \frac {(a + b)^{a + b}}{a^{a} b^{b}} \right \}^{\frac {1}{a b}}, \] wo \[ \text{Si} x = \int _{0}^{x} \frac {\sin{} u}{u} d u; \quad \text{Ci} x = \int _{\infty}^x \frac {\cos{} u}{u} d u. \]