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A sigle wave


Superposition of waves



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Dirac delta function



1. The general definition




The general definition of this function is:
δa (x - xo) = 1/2a in [xo - a , xo + a] and zero otherwise.
δ (x - xo) = lim δa (x - xo) a → 0
∫ f(x) δ(x - xo) dx = ∫ f(x) lim δa(x - xo) dx a → 0 [- ∞, + ∞]
= lim ∫ f(x) δa(x - xo) dx a → 0 [- ∞, + ∞]
= lim ∫ f(x) δa(x - xo) dx a → 0 [xo - a, xo + a]
The variable x is now taken in the interval [xo - a, xo + a], then if a → 0 then x → xo. Therefore:
∫ f(x) δ(x - xo) dx = lim (1/2a) ∫ f(xo) dx = f(xo lim (1/2a) ∫ ) dx = f(xo) lim (1/2a) [x] (between xo - a and xo + a) a → 0
= f(xo) lim (1/2a) [(xo + a) - (xo - a)] = f(xo) lim (1/2a) [2a] = f(xo). Therefore:

∫ f(x) δ(x - xo) dx = f(xo)

At the origin: xo = 0, we have the δ function &delta(x) such as:
∫ f(x) δ(x) dx = f(0).




2. Properties of the Dirac function


		
  
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