Lecture 16
t.dz?ze-IW;-zZ-zeiwolz-eiw0)(z-e-iwoJ-)
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✗ ( Z ) = ? i s u n d e f i n e d[
✗(n) = XD() §É”” e-inn
, n I w o n
The set of zs where R.s. of the sum
is the complementof the set
where the left side
converges .
e-iwn +§(éicw+n
e-” ” !=£ce”””Y+nÉ(é”””)
Digression
distribution )
A distribution maps a
function *
distribution
8 (fct) ] I f-(o )
often write
as Gct and use
the notation
of8(t) f-(t) dt E- f-(o ) ; special ease :
Can generalize b- a broader family
I[fo r ] 8.to
to ) lfctdt
Technically
However, this is true ✗
=(f) fctdt Iim [Pe
Sct) fctdt
to at )fCt)dt= /
f-(to)= Lim f Pelt-
✗a (w)= a (Scw- wo)+ 8(wtwo)
✗dcw)e”” dw= Ig/(8Cw- Wo)+scwtwoi
(won) ✗(n) =
XD(w ) = cos (won)
N- =÷e- +g-
e-icw-w.in
1- e-JCW- wo)N
1- e-Icw- wo )
I e-Jcwtwo)N +£-
1- e-Jcwtwo)
→ (w – Wo)N/z eéicw- Wo) 42
ejcw- Wo)N1z
– ICW – Wo) Nlz
ICW Wo) 112
e-Jcwtwo)N/z ejcw-w.IM/z-e-J(w-wo)N1z–ze-#-wzeJ(w-wo)Yz-e-jC)w4-z
,géj(w – Wo)”¥SinÑ”””-
sinkw-w.IN/z] centered
e-icwtwo )^¥
Sin /(wtwo) 42]
Periodic since
Éto / Xdcw) /
[(w – no)kz] /
“/ sin [(wtwo) 2]
this approaches two impulses .
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