delta_function

delta_function - DEFINITION AND PROPERTIES OF THE DIRAC...

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DEFINITION AND PROPERTIES OF THE DIRAC DELTA FUNCTION Consider the function g(t) centered on t = a (as shown below) where: g(t) = h ; - b/2 < t < b/ 2 = 0 elsewhere In addition, we will choose h such that the area under the g(t) curve is unity; i.e., we will let: h = 1/b h a b/2 b/2 g(t) t The “Dirac delta” function, δ (t-a), is defined from the above function g(t) as: δ t ( 29 = lim b 0 g(t) while maintaining hb = 1
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Properties of the Dirac delta function: i) Function value : δ t ( 29 = ∞ ; t = a = 0 t a a δ (t) t ii) Integral of δ (t) : By definition above, the area under δ (t) is unity. Therefore: δ t - a ( 29 dt 0 t = t < a = 1 t a iii) Integral of δ (t) with another function : For a function f(t) we can write (for t > a) (see following figure):
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delta_function - DEFINITION AND PROPERTIES OF THE DIRAC...

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