Appendix Aj
j Complex Variable Theory
j j
, TO jACCOMPANY
j AUTOMATIC j CONTROL j SYSTEMS
EIGHTH j EDITION
BY
j BENJAMIN jC. jKUO
j FARID
jGOLNARAGHI
JOHN j WILEY j & j SONS, j INC.
,Copyright j© j2003 jJohn jWiley j& jSons,
jInc.jAll jrights jreserved.
No jpart jof jthis jpublication jmay jbe jreproduced, jstored jin ja jretrieval jsystem jor
jtransmitted jin jany jform jor jby jany jmeans, jelectronic, jmechanical, jphotocopying,
jrecording, jscanning jor jotherwise, jexcept jas jpermitted junder jSections j107 jor j108
jof jthe j1976 jUnited jStates jCopyright jAct, jwithout jeither jthe jprior jwritten
jpermission jof jthe jPublisher, jor jauthorization jthrough jpayment jof jthe jappropriate
jper-copy jfee jto jthe jCopyright jClearance jCenter, j222 jRosewood jDrive, jDanvers,
jMA j01923, j(508)750-8400, jfax j(508)750-4470. jRequests jto jthe jPublisher jfor
jpermission jshould jbe jaddressed jto jthe jPermissions jDepartment, jJohn jWiley j&
jSons, jInc., j605 jThird jAvenue, jNew jYork, jNY j10158-0012, j(212) j850-6011, jfax
j(212) j850-6008, jE-Mail: .
To jorder jbooks jor jfor jcustomer jservice jplease jcall j1-800-CALL
jWILEY j(225-5945).
ISBN j0-471-13476-7
, ►
APPENDIX
A
Complex-Variable Theory j
► A-1 j COMPLEX-VARIABLE j CONCEPT
A-1-1 Complex j Variable
A jcomplex jvariable js jhas jtwo jcomponents: ja jreal jcomponent jσ jand jan jimaginary
jcomponent jω. jGraphically, jthe jreal jcomponent jof js jis jrepresented jby ja jσ jaxis jin
jthe jhorizontal jdirection, jand jthe jimaginary jcomponent jis jmeasured jalong jthe
jvertical jjω jaxis, jin jthe jcomplex js-plane. jFigure jA-1 jillustrates jthe jcomplex js-plane,
jin j which j any jarbitrary j point j s j = j s1 j is j defined j by j the j coordinates j σ j =
j σ1, j and j ω j = j ω1, j or j simply js1 j = jσ1 j + jjω1.
A-1-2 Functions j of j a j Complex j Variable
The j function j G(s) j is j said j to j be j a j function j of j the j complex j variable j s j if j for j every
j value j of js, jthere jis jone jor jmore jcorresponding jvalues jof jG(s). jSince js jis jdefined jto
jhave jreal jand jimaginary jparts, jthe jfunction jG(s) jis jalso jrepresented jby jits jreal jand
jimaginary jparts; jthat jis,
G1s2 j = jRe jG1s2 j + jj jIm jG1s2 (A-1)
where jRe jG(s) jdenotes jthe jreal jpart jof jG(s), jand jIm jG(s) jrepresents jthe jimaginary
jpart jof jG(s). j The j function j G(s) j is j also j represented j by j the j complex j G(s)-plane,
j with j Re j G(s) j as jthe jreal jaxis jand jIm jG(s) jas jthe jimaginary jaxis. jIf jfor jevery
jvalue jof js jthere jis jonly jone jcorresponding jvalue jof jG(s) jin jthe jG(s)-plane, jG(s) jis
jsaid jto jbe ja jsingle-valued jfunc- jtion, jand jthe jmapping jfrom jpoints jin jthe js-plane
jonto jpoints jin jthe jG(s)-plane jis jdescribed
jv
s-plane
v1 s1 s1 + jv1
0 s1 s
Figure jA-1 j The j complex j s-plane.
A-1
j Complex Variable Theory
j j
, TO jACCOMPANY
j AUTOMATIC j CONTROL j SYSTEMS
EIGHTH j EDITION
BY
j BENJAMIN jC. jKUO
j FARID
jGOLNARAGHI
JOHN j WILEY j & j SONS, j INC.
,Copyright j© j2003 jJohn jWiley j& jSons,
jInc.jAll jrights jreserved.
No jpart jof jthis jpublication jmay jbe jreproduced, jstored jin ja jretrieval jsystem jor
jtransmitted jin jany jform jor jby jany jmeans, jelectronic, jmechanical, jphotocopying,
jrecording, jscanning jor jotherwise, jexcept jas jpermitted junder jSections j107 jor j108
jof jthe j1976 jUnited jStates jCopyright jAct, jwithout jeither jthe jprior jwritten
jpermission jof jthe jPublisher, jor jauthorization jthrough jpayment jof jthe jappropriate
jper-copy jfee jto jthe jCopyright jClearance jCenter, j222 jRosewood jDrive, jDanvers,
jMA j01923, j(508)750-8400, jfax j(508)750-4470. jRequests jto jthe jPublisher jfor
jpermission jshould jbe jaddressed jto jthe jPermissions jDepartment, jJohn jWiley j&
jSons, jInc., j605 jThird jAvenue, jNew jYork, jNY j10158-0012, j(212) j850-6011, jfax
j(212) j850-6008, jE-Mail: .
To jorder jbooks jor jfor jcustomer jservice jplease jcall j1-800-CALL
jWILEY j(225-5945).
ISBN j0-471-13476-7
, ►
APPENDIX
A
Complex-Variable Theory j
► A-1 j COMPLEX-VARIABLE j CONCEPT
A-1-1 Complex j Variable
A jcomplex jvariable js jhas jtwo jcomponents: ja jreal jcomponent jσ jand jan jimaginary
jcomponent jω. jGraphically, jthe jreal jcomponent jof js jis jrepresented jby ja jσ jaxis jin
jthe jhorizontal jdirection, jand jthe jimaginary jcomponent jis jmeasured jalong jthe
jvertical jjω jaxis, jin jthe jcomplex js-plane. jFigure jA-1 jillustrates jthe jcomplex js-plane,
jin j which j any jarbitrary j point j s j = j s1 j is j defined j by j the j coordinates j σ j =
j σ1, j and j ω j = j ω1, j or j simply js1 j = jσ1 j + jjω1.
A-1-2 Functions j of j a j Complex j Variable
The j function j G(s) j is j said j to j be j a j function j of j the j complex j variable j s j if j for j every
j value j of js, jthere jis jone jor jmore jcorresponding jvalues jof jG(s). jSince js jis jdefined jto
jhave jreal jand jimaginary jparts, jthe jfunction jG(s) jis jalso jrepresented jby jits jreal jand
jimaginary jparts; jthat jis,
G1s2 j = jRe jG1s2 j + jj jIm jG1s2 (A-1)
where jRe jG(s) jdenotes jthe jreal jpart jof jG(s), jand jIm jG(s) jrepresents jthe jimaginary
jpart jof jG(s). j The j function j G(s) j is j also j represented j by j the j complex j G(s)-plane,
j with j Re j G(s) j as jthe jreal jaxis jand jIm jG(s) jas jthe jimaginary jaxis. jIf jfor jevery
jvalue jof js jthere jis jonly jone jcorresponding jvalue jof jG(s) jin jthe jG(s)-plane, jG(s) jis
jsaid jto jbe ja jsingle-valued jfunc- jtion, jand jthe jmapping jfrom jpoints jin jthe js-plane
jonto jpoints jin jthe jG(s)-plane jis jdescribed
jv
s-plane
v1 s1 s1 + jv1
0 s1 s
Figure jA-1 j The j complex j s-plane.
A-1
