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.))

If the delta function is acting at the origin, i.e., if a =0, the regularized delta function defined by (15) becomes δε(x)= 1 2ε 1+cos πx ε if −ε<x <ε, 0 otherwise. (17) Another example of regularized delta function is a sequence of bell-shaped pulses defined as δk(x−a)= 1 k √ 2π e− 1 2(x−a k) 2 (18) where k is a parameter.

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  Sequence, Delta, The delta

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