5|c|Table 1: Functions and discontinuities | ||||
function | order | A_{0} | A_{1} | A_{2} |
H(t) | 1 | |||
1 | ||||
1 | ||||
1 | 1 | |||
2 | ||||
2 | 2 | |||
1 | ||||
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 . 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) |
Definition: The function f(t) has a discontinuity of order r at the point t=0, if for all and for .Values are called amplitudes of order k.