Functions with discontinuities

Let us consider the functions listed in the left column of Table 1.

function order A₀ A₁ A₂

H(t) 1

$$e^{-\alpha t} \cdot H(t) - \alpha \alpha^2$$ 1

$$e^{-\alpha t} \sin \omega_0 t \cdot H(t) \omega_0 2 \alpha \omega_0$$ 1

$$t e^{-\alpha t} \cos \omega_0 t \cdot H(t) -2 \alpha$$ 1 1

$$t e^{-\alpha t} \sin \omega_0 t \cdot H(t) 2 \omega_0$$ 2

$$t^2 e^{-\alpha t} \cos \omega_0 t \cdot H(t)$$ 2 2

$$e^{-\alpha \vert t \vert} -2 \alpha 2 \alpha^2$$ 1

$$e^{-\alpha \vert t \vert} \sin \omega_0 t -4 \alpha \omega_0$$ 1

All these functions have one general property in common: a discontinuity (a jump of derivatives) at the point t=0. The simplest example of a discontinuity is given by the step function H(t) (Figure ). It has a discontinuity of order 0. We observe the same discontinuity for the second function $e^{- \alpha t} H(t)$. One can find the orders of the discontinuities of the other functions in column 2. In columns 3 etc., the amplitudes of derivative jumps are given:

 

A_k=f^(k)(0⁺)-f^(k)(0^-) (1)

where

$$f^{(k)}(0^{+})=\lim_{t\rightarrow 0^{+}} {\:}f^{(k)}(t),$$

$$f^{(k)}(0{-})=\lim_{t\rightarrow 0^{-}} {\:}f^{(k)}(t).$$

Definition: The function f(t) has a discontinuity of order r at the point t=0, if for all $k<r,\ A_{k}=0$ and for $k=r,\ A_{r} \neq 0$.Values $A_{k},\ (k \geq r)$ are called amplitudes of order k.