Transcription of Delta Function and Heaviside Function - IIST
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Delta Function and Heaviside Function - IIST
Delta Function and Heaviside FunctionA. SalihDepartment of Aerospace EngineeringIndian Institute of Space Science and Technology, Thiruvananthapuram 12 February 2015 We discuss some of the basic properties of the generalized functions , viz., Dirac- Delta func-tion and Heaviside step step functionThe one-dimensional Heaviside step Function centered atais defined in the following wayH(x a) =(0ifx<a,1ifx>a.(1a)Fora=0the discontinuity is atx=0, thus we haveH(x) =(0ifx<0,1ifx>0.(1b)The Heaviside Function is displayed in Fig. (x a)1a0xH(x)1 Figure 1:The Heaviside functionsH(x a)andH(x).Dirac- Delta functionTo understand the behaviour of Dirac- Delta Function (or Delta Function , for short) (x), weconsider the rectangular pulse Function (x,a) = hifa 12h<x<a+12h,0otherwise.(2)10x (x,a)haa 12ha+12hFigure 2:The pulse figure 2, it can be seen that ash , the amplitude of pulse becomes very large andits width becomes very small so that for any value ofh, the integral of the rectangular pulseZ (x,a)dx=1if the the integral of definition(a 12h,a+12h)lies in the interval( , ), and zero if range ofintegration does not contain the pulse.))
Regularized Dirac-delta function Instead of using the limit of ever-narrowing rectangular pulse of unit area when defining delta function, any similar functions can be used, provided their integral is unity and their amplitude increase as their pulse-like property narrows. For example, a regularized (smeared-out) delta
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