6.5: Impulse functions
( \newcommand{\kernel}{\mathrm{null}\,}\)
Unit impulse function = Dirac delta function is a generalized function with the properties:
- δ(t)=0,t≠0
- ∫−∞∞δ(t)dt=1
- L(δ(t−t0))=e−st0
Since the first item is just a definition we will prove 2. and 3.
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.
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
The following are some useful identities:
- sin(t)=eit−e−it2i
- cos(t)=eit+e−it2
- sinh(t)=et−e−t2
- cosh(t)=et+e−t2
- [sinh(t)]′=cosh(t)
- [cosh(t)]′=sinh(t)
- sinh(0)=e0−e02=0
- 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)