StepFunctions - Step Functions To deal effectively with...

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Step Functions To deal effectively with functions having jump discontinuities, it is very helpful to introduce a function known as the unit step function or Heaviside function . This function is denoted by u c where c 0, and it is defined by: u c (t) = 0 if t < c 1 if t > c ! " # When c = 0, we have u 0 (t) = 1 Notice u 0 (t) ! u c (t) = 1 if 0 < t < c 0 if t > c " # $
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A variety of so-called step functions can be expressed as a linear combination of unit step functions. Example : 1) f(t) = 0 if 0 < t < 3 5 if t > 3 = 5 0 if 0 < t < 3 1 if t > 3 = 5u 3 (t) ! " # ! " # 2) g(t) = 3 if 0 < t < 7 9 if t > 7 ! " # = 3 + 0 if 0 < t < 7 6 if t > 7 = 3 + 6 0 if 0 < t < 7 1 if t > 7 ! " # ! " # = 3u 0 (t) + 6u 7 (t) 3) h(t) = 0 if 0 < t < 1 2 if 1 < t < 3 1 if t > 3 = 0 if 0 < t < 1 2 if t > 1 + 0 if 0 < t < 3 ! 1 if t > 3 = " # $ " # $ " # % $ % 2 0 if 0 < t < 1 1 if t > 1 ! 0 if 0 < t < 3 1 if t > 3 = " # $ " # $ 2u 1 (t) ! u 3 (t) 4) p(t) = 2 if 0 < t < 3 ! 3 if 3 < t < 5 6 if t > 5 = 2 " # $ % $ + 0 if 0 < t < 3 ! 5 if 3 < t < 5 4 if t > 5 = 2 + 0 if 0 < t < 3 ! 5 if t
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StepFunctions - Step Functions To deal effectively with...

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