Chapter 11.10, Problem 4P

$E_n(x)={\displaystyle {\int }_1^{\infty }\frac{e^{-xt}}{t^n}}dt,n=0,1,2,\cdots ,$ and $\text{Ei}(x)={\displaystyle {\int }_{-\infty }^x\frac{e^t}{t}}dt,$ and other similar integrals are called exponential integrals. By making appropriate changes of variable, show that

(a) $E_1(x)={\displaystyle {\int }_x^{\infty }\frac{e^{-t}}{t}}dt$

(b) $\text{Ei}(x)=-{\displaystyle {\int }_{-\infty }^{\infty }\frac{e^{-t}}{t}}dt$

(c) $E_1(x)=-\text{Ei}(-x)$

(d) ${\displaystyle {\int }_0^x\frac{e^{1/t}}{t}}dt=E_1(-1/x)$

(Caution: Various notations are used; check carefully the notation in references you are using.)