No~ thqf l(~S middot-IOtL

is v-J)~ ~5ect Jo uJ o~ middotu 1 ~-ctJL

5J l ~ d JL- c j i c~ J 0 rek5jto orr rdJu R i fkshy

_______--~-------~(rrr-----~--(~lT--- of CO1v~C~ re - Goi~ tlftou~ k

CltYIkuh OIC~ lt(Qvt V cJocklv I~ IrCfpound~ Lu Ro Iyenshy

---0 to 217 0cUt

~ b((Yt~ Q0 oi uobJ---- 1 1Vwshypoundpound4ji~ G) 5 Igt ot 2--i l= LJc~~kS o

+v-c-f5figtI ~f--shy

AR~ncd-I-~ IV V-VSflt ~ ~ -v-C(s Q r fYL ~~ reg

a JJ (11) ~_z~7 _1_ t gt0(I-CLi- (

(cS

(d JiVmiddot io)

3 COvq~ V ~O~ bull

Residues The contour integral can be evaluated using standard techniques from complex variable theory There is a theorem due to Caehy which says that if J(z) is single-valued and analytic on and inside a dosed contour C except for a finite nwnber of poles PI P2 pN inside C then

1 N (7)21fmiddot i J(z)dz = L Rk J C k=l

where Hk is lht so-called residue of f() al Lhr pole p When PI is a simpl pole (as against a multiple pole such as a double pole) t lie l the residlle ilt calc1llated IIS follows

(8)

If Pk is a pole of order M then the residue is calculated using a more complica~ed formula

1 d M

-I

[ ]Rk = Z~~k (M _ I) dz lvI - 1 (z - Pk)M J(z) (9)

So given a rational function X(z) aml H specified region of convergence HI lt Izl lt R 2 we can compute Ihe inverse z-transform [(n) as follows

1 I[Rke R liRt of tlle r o1es of Z- l X(z) in the region 0 Izi lt Rl middot

2 CompuLc the residues of Lhese poles and add them up The result is precisciy I(n)

This method is called the contowmiddot integration method for computing the inverse z-transform It can be somewhat cumbersome because of the poles created at z = 0 by the factor zn-l in Eq (6) (when n lt -1) These arc multiple poles and the multiplicity n -1 depends on the time index n Calculating the residues for these can be laborious But there are examples where the contour integration method is the most convenient For example problem 339 in AVOs book is easy to solve using contour integration

~ vrF IlfN~ ~hod ~ i~2J to eXres s X(~) C0 ~

L (Au ( eurof t st2f I ES ~

X(-2-) =- ~

J([~1 irl-

Iv) Ro c ~ ~ I - i l I lt I~I lt 00

L~ure~ Ss- i~

t- ~ ROC

~ 5l +

I - l- I

Cc- f o t v~~ eA(=-Al3 a~

S f(~ l-t gt IcMFmiddott~ i ( tNL lZoe

-1 0 D~ 1 0 Lauf~l-+ 3Qri~ -exrogtDo--- ~(X(~) ~

veuro- (Ji riL X C~) ~s

S lCl2 I~ l ( II ~~ ~~ 1 ~+i~

o

)( (t-) ~ ~ 1 ( I -t r- + ~-+ cgt-+ )

Xc~) =- - (-l +r--+ -t Zc + -)

XCZ) - L06

9shy

(~

n = -cgt

~ 0ok~~ C[I ] ~ ~ (

0C CI) J = 0

3 37 -+ 3

10

+1 = _J ~_ oIIfL6 shy

-2 3

--- - 2

( 5) -(- 3) -= l =- - 5

F Cl-) = (z- +3 )

~ ( t+2) Z- (z- +5)

-t --Z+S

Most- ot ~ UV~OvvA coJ+iC~~gt C~ ~ rbucL ~ ~ cou-V ~oJj ~o cL _

2-4-

~s (jJ~J kJ ~ ~

-+ ~f-) A~ + ~T~ AJ ( t2) ~ t9 2shy

~ ( z ~ s) -- (~t3) ( 2l +s) ~(~ t-5~ 2

I N I J - bull bull N( ~

( -==0

k=

0( I J J 01N D V -Z roQs o X(7)

LU-t ~ MveuroobpLi clJ cf ro~ a--(- os N~

ik S I fVes-r (~ IS W~~ ~ -po)QS (A~- XC) QV~

ol Sj Mr--lt- (no pol~ IS (~re=k~

t I ~ of 11Q frln-I k X Itt ) lA-I

I k -fU(ctio- X(~) 2vcc1-IQI S ~s

- - 1- O( i shyx(~) shy

If o-~ rO-lL ~s (~~ (Y~ ih~f alc-e II X C~ J ~ ~ rcr-t cJL fl~ 01 ~7c $ 10 -- Loftoir-5gt (ore ~fl~

Tal fS--c-A C ~ ) f faZ eX -( ( s r -euro ~~k~ Iv i f-i YL6 ~

ycr1i~ flfCAd-lOf bcASTOI of )((1-) ~I C91~il 1k~tY-S

f ~ 1- fIttLl ~ - c~ i~ - -~-0~12 )-l -0(

I 1-fV-Jl- v+ ~o~ ~ i c~) c 1fL 0( ) - OltN

~h- MuQ+ UC I~~ -of l~ 0ltlt ~~ N ~ (FE

ot 1lt-) W iI bL

tJ ~

X(~) = LrL~ (=- k (

2

A- +

-fi~ A ~l Jgt- MdtiS beth s t-s 6 (( ~i )( 1- Lpound) -= AI (1- lt2i) -+ AL ((-~)

I - ~ _ rf J - IQLA cd- l l NL lt c I ytc tCJ vV) Cl -c f Q CO q-T i C Q1lb Of L

vJ C2 r ~= ~LA cL- L=---- 5gt o~ tw= J A l Agt VJ fs

f- ~ ~ ft2- ~ L

_ 2-A( - A2 = O

iI lt I

( lt r 2

I =L 1 )

-+ ~ ( -(2f4 [-n-V (-+ 1- 0) l-n-JM

RoC 2- lt I=l- L lt 0lt)

ral~(2 _ C-~ 2

pJ~ cd ~ lj i ~

~ ya~ 4- -1 he-s ~~Iclld 2

~twj boh s ~ IJ ~~ to~y- (1- 2- lti=)

~_ 4lti) =- 1+11 C1- i Xi - 2i) ~ All ( 1 ~ Z -i) -+~ (( - ij - Ao 1 1 (1 - 3 i +2i) -+ -4 I2 ( - 2- i -I-Az1 ( 1~ 2 i

shy

f I I ~ A 2-f A2-1 ~ I

41 Zshy

AIL ~ 3

421 ~ ~~

3

X(~) I-it - [li

)--00u12- 2r1elL GUA5 ~ C _1Yc-~hrM-

( - o( i- I w~ Icrtv v-gt Ikcct

5 0lt( v LAj

10 f ld- i A~ ~-t-r-C- IS ~(~ cl uSQL n ~ ( ) E ~ ~ Z cL X(-l)

J-l

AR~ncd-I-~ IV V-VSflt ~ ~ -v-C(s Q r fYL ~~ reg

a JJ (11) ~_z~7 _1_ t gt0(I-CLi- (

(cS

(d JiVmiddot io)

3 COvq~ V ~O~ bull

Residues The contour integral can be evaluated using standard techniques from complex variable theory There is a theorem due to Caehy which says that if J(z) is single-valued and analytic on and inside a dosed contour C except for a finite nwnber of poles PI P2 pN inside C then

1 N (7)21fmiddot i J(z)dz = L Rk J C k=l

where Hk is lht so-called residue of f() al Lhr pole p When PI is a simpl pole (as against a multiple pole such as a double pole) t lie l the residlle ilt calc1llated IIS follows

(8)

If Pk is a pole of order M then the residue is calculated using a more complica~ed formula

1 d M

-I

[ ]Rk = Z~~k (M _ I) dz lvI - 1 (z - Pk)M J(z) (9)

So given a rational function X(z) aml H specified region of convergence HI lt Izl lt R 2 we can compute Ihe inverse z-transform [(n) as follows

1 I[Rke R liRt of tlle r o1es of Z- l X(z) in the region 0 Izi lt Rl middot

2 CompuLc the residues of Lhese poles and add them up The result is precisciy I(n)

This method is called the contowmiddot integration method for computing the inverse z-transform It can be somewhat cumbersome because of the poles created at z = 0 by the factor zn-l in Eq (6) (when n lt -1) These arc multiple poles and the multiplicity n -1 depends on the time index n Calculating the residues for these can be laborious But there are examples where the contour integration method is the most convenient For example problem 339 in AVOs book is easy to solve using contour integration

~ vrF IlfN~ ~hod ~ i~2J to eXres s X(~) C0 ~

L (Au ( eurof t st2f I ES ~

X(-2-) =- ~

J([~1 irl-

Iv) Ro c ~ ~ I - i l I lt I~I lt 00

L~ure~ Ss- i~

t- ~ ROC

~ 5l +

I - l- I

Cc- f o t v~~ eA(=-Al3 a~

S f(~ l-t gt IcMFmiddott~ i ( tNL lZoe

-1 0 D~ 1 0 Lauf~l-+ 3Qri~ -exrogtDo--- ~(X(~) ~

veuro- (Ji riL X C~) ~s

S lCl2 I~ l ( II ~~ ~~ 1 ~+i~

o

)( (t-) ~ ~ 1 ( I -t r- + ~-+ cgt-+ )

Xc~) =- - (-l +r--+ -t Zc + -)

XCZ) - L06

9shy

(~

n = -cgt

~ 0ok~~ C[I ] ~ ~ (

0C CI) J = 0

3 37 -+ 3

10

+1 = _J ~_ oIIfL6 shy

-2 3

--- - 2

( 5) -(- 3) -= l =- - 5

F Cl-) = (z- +3 )

~ ( t+2) Z- (z- +5)

-t --Z+S

Most- ot ~ UV~OvvA coJ+iC~~gt C~ ~ rbucL ~ ~ cou-V ~oJj ~o cL _

2-4-

~s (jJ~J kJ ~ ~

-+ ~f-) A~ + ~T~ AJ ( t2) ~ t9 2shy

~ ( z ~ s) -- (~t3) ( 2l +s) ~(~ t-5~ 2

I N I J - bull bull N( ~

( -==0

k=

0( I J J 01N D V -Z roQs o X(7)

LU-t ~ MveuroobpLi clJ cf ro~ a--(- os N~

ik S I fVes-r (~ IS W~~ ~ -po)QS (A~- XC) QV~

ol Sj Mr--lt- (no pol~ IS (~re=k~

t I ~ of 11Q frln-I k X Itt ) lA-I

I k -fU(ctio- X(~) 2vcc1-IQI S ~s

- - 1- O( i shyx(~) shy

If o-~ rO-lL ~s (~~ (Y~ ih~f alc-e II X C~ J ~ ~ rcr-t cJL fl~ 01 ~7c $ 10 -- Loftoir-5gt (ore ~fl~

Tal fS--c-A C ~ ) f faZ eX -( ( s r -euro ~~k~ Iv i f-i YL6 ~

ycr1i~ flfCAd-lOf bcASTOI of )((1-) ~I C91~il 1k~tY-S

f ~ 1- fIttLl ~ - c~ i~ - -~-0~12 )-l -0(

I 1-fV-Jl- v+ ~o~ ~ i c~) c 1fL 0( ) - OltN

~h- MuQ+ UC I~~ -of l~ 0ltlt ~~ N ~ (FE

ot 1lt-) W iI bL

tJ ~

X(~) = LrL~ (=- k (

2

A- +

-fi~ A ~l Jgt- MdtiS beth s t-s 6 (( ~i )( 1- Lpound) -= AI (1- lt2i) -+ AL ((-~)

I - ~ _ rf J - IQLA cd- l l NL lt c I ytc tCJ vV) Cl -c f Q CO q-T i C Q1lb Of L

vJ C2 r ~= ~LA cL- L=---- 5gt o~ tw= J A l Agt VJ fs

f- ~ ~ ft2- ~ L

_ 2-A( - A2 = O

iI lt I

( lt r 2

I =L 1 )

-+ ~ ( -(2f4 [-n-V (-+ 1- 0) l-n-JM

RoC 2- lt I=l- L lt 0lt)

ral~(2 _ C-~ 2

pJ~ cd ~ lj i ~

~ ya~ 4- -1 he-s ~~Iclld 2

~twj boh s ~ IJ ~~ to~y- (1- 2- lti=)

~_ 4lti) =- 1+11 C1- i Xi - 2i) ~ All ( 1 ~ Z -i) -+~ (( - ij - Ao 1 1 (1 - 3 i +2i) -+ -4 I2 ( - 2- i -I-Az1 ( 1~ 2 i

shy

f I I ~ A 2-f A2-1 ~ I

41 Zshy

AIL ~ 3

421 ~ ~~

3

X(~) I-it - [li

)--00u12- 2r1elL GUA5 ~ C _1Yc-~hrM-

( - o( i- I w~ Icrtv v-gt Ikcct

5 0lt( v LAj

10 f ld- i A~ ~-t-r-C- IS ~(~ cl uSQL n ~ ( ) E ~ ~ Z cL X(-l)

J-l

Residues The contour integral can be evaluated using standard techniques from complex variable theory There is a theorem due to Caehy which says that if J(z) is single-valued and analytic on and inside a dosed contour C except for a finite nwnber of poles PI P2 pN inside C then

1 N (7)21fmiddot i J(z)dz = L Rk J C k=l

where Hk is lht so-called residue of f() al Lhr pole p When PI is a simpl pole (as against a multiple pole such as a double pole) t lie l the residlle ilt calc1llated IIS follows

(8)

If Pk is a pole of order M then the residue is calculated using a more complica~ed formula

1 d M

-I

[ ]Rk = Z~~k (M _ I) dz lvI - 1 (z - Pk)M J(z) (9)

So given a rational function X(z) aml H specified region of convergence HI lt Izl lt R 2 we can compute Ihe inverse z-transform [(n) as follows

1 I[Rke R liRt of tlle r o1es of Z- l X(z) in the region 0 Izi lt Rl middot

2 CompuLc the residues of Lhese poles and add them up The result is precisciy I(n)

This method is called the contowmiddot integration method for computing the inverse z-transform It can be somewhat cumbersome because of the poles created at z = 0 by the factor zn-l in Eq (6) (when n lt -1) These arc multiple poles and the multiplicity n -1 depends on the time index n Calculating the residues for these can be laborious But there are examples where the contour integration method is the most convenient For example problem 339 in AVOs book is easy to solve using contour integration

~ vrF IlfN~ ~hod ~ i~2J to eXres s X(~) C0 ~

L (Au ( eurof t st2f I ES ~

X(-2-) =- ~

J([~1 irl-

Iv) Ro c ~ ~ I - i l I lt I~I lt 00

L~ure~ Ss- i~

t- ~ ROC

~ 5l +

I - l- I

Cc- f o t v~~ eA(=-Al3 a~

S f(~ l-t gt IcMFmiddott~ i ( tNL lZoe

-1 0 D~ 1 0 Lauf~l-+ 3Qri~ -exrogtDo--- ~(X(~) ~

veuro- (Ji riL X C~) ~s

S lCl2 I~ l ( II ~~ ~~ 1 ~+i~

o

)( (t-) ~ ~ 1 ( I -t r- + ~-+ cgt-+ )

Xc~) =- - (-l +r--+ -t Zc + -)

XCZ) - L06

9shy

(~

n = -cgt

~ 0ok~~ C[I ] ~ ~ (

0C CI) J = 0

3 37 -+ 3

10

+1 = _J ~_ oIIfL6 shy

-2 3

--- - 2

( 5) -(- 3) -= l =- - 5

F Cl-) = (z- +3 )

~ ( t+2) Z- (z- +5)

-t --Z+S

Most- ot ~ UV~OvvA coJ+iC~~gt C~ ~ rbucL ~ ~ cou-V ~oJj ~o cL _

2-4-

~s (jJ~J kJ ~ ~

-+ ~f-) A~ + ~T~ AJ ( t2) ~ t9 2shy

~ ( z ~ s) -- (~t3) ( 2l +s) ~(~ t-5~ 2

I N I J - bull bull N( ~

( -==0

k=

0( I J J 01N D V -Z roQs o X(7)

LU-t ~ MveuroobpLi clJ cf ro~ a--(- os N~

ik S I fVes-r (~ IS W~~ ~ -po)QS (A~- XC) QV~

ol Sj Mr--lt- (no pol~ IS (~re=k~

t I ~ of 11Q frln-I k X Itt ) lA-I

I k -fU(ctio- X(~) 2vcc1-IQI S ~s

- - 1- O( i shyx(~) shy

If o-~ rO-lL ~s (~~ (Y~ ih~f alc-e II X C~ J ~ ~ rcr-t cJL fl~ 01 ~7c $ 10 -- Loftoir-5gt (ore ~fl~

Tal fS--c-A C ~ ) f faZ eX -( ( s r -euro ~~k~ Iv i f-i YL6 ~

ycr1i~ flfCAd-lOf bcASTOI of )((1-) ~I C91~il 1k~tY-S

f ~ 1- fIttLl ~ - c~ i~ - -~-0~12 )-l -0(

I 1-fV-Jl- v+ ~o~ ~ i c~) c 1fL 0( ) - OltN

~h- MuQ+ UC I~~ -of l~ 0ltlt ~~ N ~ (FE

ot 1lt-) W iI bL

tJ ~

X(~) = LrL~ (=- k (

2

A- +

-fi~ A ~l Jgt- MdtiS beth s t-s 6 (( ~i )( 1- Lpound) -= AI (1- lt2i) -+ AL ((-~)

I - ~ _ rf J - IQLA cd- l l NL lt c I ytc tCJ vV) Cl -c f Q CO q-T i C Q1lb Of L

vJ C2 r ~= ~LA cL- L=---- 5gt o~ tw= J A l Agt VJ fs

f- ~ ~ ft2- ~ L

_ 2-A( - A2 = O

iI lt I

( lt r 2

I =L 1 )

-+ ~ ( -(2f4 [-n-V (-+ 1- 0) l-n-JM

RoC 2- lt I=l- L lt 0lt)

ral~(2 _ C-~ 2

pJ~ cd ~ lj i ~

~ ya~ 4- -1 he-s ~~Iclld 2

~twj boh s ~ IJ ~~ to~y- (1- 2- lti=)

~_ 4lti) =- 1+11 C1- i Xi - 2i) ~ All ( 1 ~ Z -i) -+~ (( - ij - Ao 1 1 (1 - 3 i +2i) -+ -4 I2 ( - 2- i -I-Az1 ( 1~ 2 i

shy

f I I ~ A 2-f A2-1 ~ I

41 Zshy

AIL ~ 3

421 ~ ~~

3

X(~) I-it - [li

)--00u12- 2r1elL GUA5 ~ C _1Yc-~hrM-

( - o( i- I w~ Icrtv v-gt Ikcct

5 0lt( v LAj

10 f ld- i A~ ~-t-r-C- IS ~(~ cl uSQL n ~ ( ) E ~ ~ Z cL X(-l)

J-l

~ vrF IlfN~ ~hod ~ i~2J to eXres s X(~) C0 ~

L (Au ( eurof t st2f I ES ~

X(-2-) =- ~

J([~1 irl-

Iv) Ro c ~ ~ I - i l I lt I~I lt 00

L~ure~ Ss- i~

t- ~ ROC

~ 5l +

I - l- I

Cc- f o t v~~ eA(=-Al3 a~

S f(~ l-t gt IcMFmiddott~ i ( tNL lZoe

-1 0 D~ 1 0 Lauf~l-+ 3Qri~ -exrogtDo--- ~(X(~) ~

veuro- (Ji riL X C~) ~s

S lCl2 I~ l ( II ~~ ~~ 1 ~+i~

o

)( (t-) ~ ~ 1 ( I -t r- + ~-+ cgt-+ )

Xc~) =- - (-l +r--+ -t Zc + -)

XCZ) - L06

9shy

(~

n = -cgt

~ 0ok~~ C[I ] ~ ~ (

0C CI) J = 0

3 37 -+ 3

10

+1 = _J ~_ oIIfL6 shy

-2 3

--- - 2

( 5) -(- 3) -= l =- - 5

F Cl-) = (z- +3 )

~ ( t+2) Z- (z- +5)

-t --Z+S

Most- ot ~ UV~OvvA coJ+iC~~gt C~ ~ rbucL ~ ~ cou-V ~oJj ~o cL _

2-4-

~s (jJ~J kJ ~ ~

-+ ~f-) A~ + ~T~ AJ ( t2) ~ t9 2shy

~ ( z ~ s) -- (~t3) ( 2l +s) ~(~ t-5~ 2

I N I J - bull bull N( ~

( -==0

k=

0( I J J 01N D V -Z roQs o X(7)

LU-t ~ MveuroobpLi clJ cf ro~ a--(- os N~

ik S I fVes-r (~ IS W~~ ~ -po)QS (A~- XC) QV~

ol Sj Mr--lt- (no pol~ IS (~re=k~

t I ~ of 11Q frln-I k X Itt ) lA-I

I k -fU(ctio- X(~) 2vcc1-IQI S ~s

- - 1- O( i shyx(~) shy

If o-~ rO-lL ~s (~~ (Y~ ih~f alc-e II X C~ J ~ ~ rcr-t cJL fl~ 01 ~7c $ 10 -- Loftoir-5gt (ore ~fl~

Tal fS--c-A C ~ ) f faZ eX -( ( s r -euro ~~k~ Iv i f-i YL6 ~

ycr1i~ flfCAd-lOf bcASTOI of )((1-) ~I C91~il 1k~tY-S

f ~ 1- fIttLl ~ - c~ i~ - -~-0~12 )-l -0(

I 1-fV-Jl- v+ ~o~ ~ i c~) c 1fL 0( ) - OltN

~h- MuQ+ UC I~~ -of l~ 0ltlt ~~ N ~ (FE

ot 1lt-) W iI bL

tJ ~

X(~) = LrL~ (=- k (

2

A- +

-fi~ A ~l Jgt- MdtiS beth s t-s 6 (( ~i )( 1- Lpound) -= AI (1- lt2i) -+ AL ((-~)

I - ~ _ rf J - IQLA cd- l l NL lt c I ytc tCJ vV) Cl -c f Q CO q-T i C Q1lb Of L

vJ C2 r ~= ~LA cL- L=---- 5gt o~ tw= J A l Agt VJ fs

f- ~ ~ ft2- ~ L

_ 2-A( - A2 = O

iI lt I

( lt r 2

I =L 1 )

-+ ~ ( -(2f4 [-n-V (-+ 1- 0) l-n-JM

RoC 2- lt I=l- L lt 0lt)

ral~(2 _ C-~ 2

pJ~ cd ~ lj i ~

~ ya~ 4- -1 he-s ~~Iclld 2

~twj boh s ~ IJ ~~ to~y- (1- 2- lti=)

~_ 4lti) =- 1+11 C1- i Xi - 2i) ~ All ( 1 ~ Z -i) -+~ (( - ij - Ao 1 1 (1 - 3 i +2i) -+ -4 I2 ( - 2- i -I-Az1 ( 1~ 2 i

shy

f I I ~ A 2-f A2-1 ~ I

41 Zshy

AIL ~ 3

421 ~ ~~

3

X(~) I-it - [li

)--00u12- 2r1elL GUA5 ~ C _1Yc-~hrM-

( - o( i- I w~ Icrtv v-gt Ikcct

5 0lt( v LAj

10 f ld- i A~ ~-t-r-C- IS ~(~ cl uSQL n ~ ( ) E ~ ~ Z cL X(-l)

J-l

o

)( (t-) ~ ~ 1 ( I -t r- + ~-+ cgt-+ )

Xc~) =- - (-l +r--+ -t Zc + -)

XCZ) - L06

9shy

(~

n = -cgt

~ 0ok~~ C[I ] ~ ~ (

0C CI) J = 0

3 37 -+ 3

10

+1 = _J ~_ oIIfL6 shy

-2 3

--- - 2

( 5) -(- 3) -= l =- - 5

F Cl-) = (z- +3 )

~ ( t+2) Z- (z- +5)

-t --Z+S

Most- ot ~ UV~OvvA coJ+iC~~gt C~ ~ rbucL ~ ~ cou-V ~oJj ~o cL _

2-4-

~s (jJ~J kJ ~ ~

-+ ~f-) A~ + ~T~ AJ ( t2) ~ t9 2shy

~ ( z ~ s) -- (~t3) ( 2l +s) ~(~ t-5~ 2

I N I J - bull bull N( ~

( -==0

k=

0( I J J 01N D V -Z roQs o X(7)

LU-t ~ MveuroobpLi clJ cf ro~ a--(- os N~

ik S I fVes-r (~ IS W~~ ~ -po)QS (A~- XC) QV~

ol Sj Mr--lt- (no pol~ IS (~re=k~

t I ~ of 11Q frln-I k X Itt ) lA-I

I k -fU(ctio- X(~) 2vcc1-IQI S ~s

- - 1- O( i shyx(~) shy

If o-~ rO-lL ~s (~~ (Y~ ih~f alc-e II X C~ J ~ ~ rcr-t cJL fl~ 01 ~7c $ 10 -- Loftoir-5gt (ore ~fl~

Tal fS--c-A C ~ ) f faZ eX -( ( s r -euro ~~k~ Iv i f-i YL6 ~

ycr1i~ flfCAd-lOf bcASTOI of )((1-) ~I C91~il 1k~tY-S

f ~ 1- fIttLl ~ - c~ i~ - -~-0~12 )-l -0(

I 1-fV-Jl- v+ ~o~ ~ i c~) c 1fL 0( ) - OltN

~h- MuQ+ UC I~~ -of l~ 0ltlt ~~ N ~ (FE

ot 1lt-) W iI bL

tJ ~

X(~) = LrL~ (=- k (

2

A- +

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