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Οι σημειώσεις μου από το μάθημα των διακριτών μαθηματικών. Ελπίζω να βγάλετε άκρη... Σύνολα, σχέσεις, αρχή του περιστερώνα, μαθηματική επαγωγή, προτασιακή λογική, συνδυαστική, γεωμετρική πρόοδος, ακολουθίες, διωνυμικό ανάπτυγμα, γεννήτριες και αναδρομικές σχέσεις.
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~____ +-____~~~~~~~~~ __~~OamiddotAn~~=u-_______________ ~~____~____~-~~~ ________7~_~fl~~p-_____________________
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~
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caoo 61afPftI tAvtampU1~-B)hv5ul6t D y- Q~ ~f~V bull bull
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gt( t ( 1middot 10-1 --gt jsmiddot 3 90c ~ -= 3 (lttomiddot ~
~
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00
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--
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bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
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LI ~
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1- 1 10
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-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
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o
JA x61= fAl-Il)
IAU~ f lA J t IBf~ IAn B1 - t9t ~fI t va E ~r~IG---___~
A~middotr 1Jrca-10~
hArol~ 10 TI 8~amp~ nxl)
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-atCf 0) -lOt q~t~t~
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pc ~1Qltmiddote1~~ fit It
~1fo1~ Jdt1( t ~) 00 t (- gtj itDI It ~II) 16 it bull bull bull (4 shy
gt( t ( 1middot 10-1 --gt jsmiddot 3 90c ~ -= 3 (lttomiddot ~
~
AAlClt~I~ AvtIlAEvw middot (Awfffvt rt or~t~ 0foifIJV oitlUflff(IIIY)
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bullbull f
10 dfl 0 lt0rotl ~gt TJOfio fiJ1J
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caoo 61afPftI tAvtampU1~-B)hv5ul6t D y- Q~ ~f~V bull bull
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bull bull amp bull bull -
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I bull
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bull 1 bull I
((1)= 6 IIbull n- 1 r
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laquoI bull bullbull
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QttoAo94middotAS fi1 =t 4 bull bull
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bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
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lufJ t (J v Av L
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QttoAo94middotAS fi1 =t 4 bull bull
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1- 1 10
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LIe 0 A (z)= -shy1-10z 3 n
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bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
I
1 5151 2
cJ (4) 1 ( 5) 2 ri- + 1 z 2 2 f ~
middot l~ 1~ ~middot t) W1rT~v 1 ~~t 4 en 7 ~ao1 ( ~~ ~2 ) ~ ArzJ f f (3 Cz))r lt
00
~ 7 f 2 0 01 Z m 11 )
K Al2
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
1 5151 2
cJ (4) 1 ( 5) 2 ri- + 1 z 2 2 f ~
middot l~ 1~ ~middot t) W1rT~v 1 ~~t 4 en 7 ~ao1 ( ~~ ~2 ) ~ ArzJ f f (3 Cz))r lt
00
~ 7 f 2 0 01 Z m 11 )
K Al2
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
middot l~ 1~ ~middot t) W1rT~v 1 ~~t 4 en 7 ~ao1 ( ~~ ~2 ) ~ ArzJ f f (3 Cz))r lt
00
~ 7 f 2 0 01 Z m 11 )
K Al2
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I