71 Remarks on some functions

(i) In the above table $x^{\alpha }$ satisfies (E) for $\alpha \ge 0$; for $0>\alpha >-1$, $x^{\alpha }$ diverges as $x\to 0+$, but the Laplace transform integral exists as an improper Riemann integral.

(ii) The Dirac delta function ${\delta }_b$ is not actually a function, but the measure that assigns mass one to the point $b\ge 0$ on the line. So $\int f(x){\delta }_b(dx)=f(b)$ for all continuous real functions $f$.

(iii) The Heaviside function

$$H(x)=1,x\ge 0;$$

is a step function with a jump at $x=0$, so $H(x-b)$ is a step function with a jump at $x=b$. Hence $H(x-b)={\int }_{(-\mathrm{\infty },x]}{\delta }_b(dt).$