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6.5: Impulse functions

( \newcommand{\kernel}{\mathrm{null}\,}\)

Unit impulse function = Dirac delta function is a generalized function with the properties:

  1. δ(t)=0,t≠0
  2. ∫−∞∞δ(t)dt=1
  3. L(δ(t−t0))=e−st0

Since the first item is just a definition we will prove 2. and 3.

Proof of 2.

We let dk(t)={12k−k<t<k0t≤−k0t≥−k

We note that limk→0dk(t)=0 if t≠0 and limk→0∫∫−∫dk(t)=limk→01=1=∫∞−∞δ(t)dt.

Proof of 3.

L(δ(t−t0))=limk→0+L(dk(t−t0))=limk→0∫∞0e−st(dk(t−t0))dt=limk→012k∫t0+kt0−ke−stdt=limk→0−12ske−st|t0+kt0−k=limk→012ske−st0(esk−e−sk)=limk→0sinh(sk)ske−st0=limk→0scosh(sk)se−st0=e−st0

Note

The following are some useful identities:

  1. sin(t)=eit−e−it2i
  2. cos(t)=eit+e−it2
  3. sinh(t)=et−e−t2
  4. cosh(t)=et+e−t2
  5. [sinh(t)]′=cosh(t)
  6. [cosh(t)]′=sinh(t)
  7. sinh(0)=e0−e02=0
  8. cosh(0)=e0+e02=1

Intro to Group Theory
Define the â‹… product on R2 by
(x1,y1)⋅(x2,y2)=(x1x2−y1y2,x1y2−y1x2)

Note â‹… is
1.) commutative:(x1,y1)⋅(x2,y2)=(x1x2−y1y2,x1y2−y1x2)=(x2x1−y2y1,x2y1−y2x1)=(x2,y2)⋅(x1,y1)
2.) associative: (fâ‹…g)â‹…h=fâ‹…(gâ‹…h)
3.) distributive w.r.t +:fâ‹…(g1+g2)=fâ‹…g1+fâ‹…g2
4.) (x1,y1)â‹…(0,0)=(0,0)

Note: (0,1)⋅(0,1)=(−1,0)

 


This page titled 6.5: Impulse functions is shared under a not declared license and was authored, remixed, and/or curated by Isabel K. Darcy.

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