Table Of ContentTRANSFORM MATHEMATICS
OrhanÖzhan
2
˙ ˙
GIRIS¸
Bismillahirrahmanirrahim
Matematik... Onunla ilgili söylenmis¸ çok s¸ey var s¸üphesiz. Söylenenlerin en güzeli, sözlerin en güzelinde söylen-
mis¸: "Günes¸ ve ay hesap iledir1", "... ve geceyi sükunet için, günes¸ ve ayı da hesap için yarattı. I˙s¸te bu aziz olan,
her s¸eyi kemali ile bilen Allahın takdiridir2". Fiziksel evrende bilime medar olan her tabii hadise matematik ile ifade
edilebilmektedir. Evrendemaddeyleilgiliolanyaratılmıs¸hers¸eybudilleanlas¸ılıyor. Galileo’nundadedig˘igibi: "Kainat
dedig˘imizkitap,yazıldıg˘ıdilveharflerög˘renilmedikçeokunamaz. O,matematikdilindeyazılmıs¸tır;harfleriüçgen,daire
vedig˘ergeometriks¸ekillerdir. Budilveharflerolmaksızınkitabınbirtekkelimesinianlamayabes¸eriolarakimkanyok-
tur.3"Matematig˘ininsanlartarafındanyapıldıg˘ıiddiasınınaksine,buvarlıklarınkendilerihernekadarezeliolmasalarda,
onlarındiliolanmatematikAllah’ınilmidir;Allahnasılkadimisematematikdekadimdir,ezelidir. Buitibarlamatematik
insana ayrı bir haz ve tatmin verir; onun tadı ve güzellig˘i akıl ile idrak edilir. "O’nun istemesi dıs¸ında onun ilminden
hiçbirs¸eyiidrakedemezler4"ilahikelamıgereg˘ince,eg˘erbizmatematikbiliyorsakbutamamenAllah’ınlutfuveikramı
iledir; vebuinsaniçinbüyükbirs¸erefvenimettir. ˙Insanlıkvebilimtarihindekazanılmıs¸ olanbütünmatematikmükte-
sebatı kütüphaneler doldursa da, yine de bu müktesebat Allah’ın ilmi yanında okyanusdan bir damla gibidir veya daha
azdır.
˙Is¸tebuküçükmüktesebatlabugünatomunyapısıvetemelparçacıklaranlas¸ılmıs¸;astronomiilmigünes¸sistemidıs¸ına
araçgönderecek,Mars’aaraçindirecekseviyeyegelmis¸;elektronikvehaberles¸meteknolojisiakıllıtelefonlarlainsanları
yüz yüze görüs¸ür hale getirmis¸; hava ve deniz araçları ile dünya küçük ve global bir köye dönüs¸müs¸; bilgisayar ve
internet teknolojisi ile bilgi sistemlerinde adeta bir patlama yas¸anmıs¸; tıpta inanılmaz görüntüleme ve tedavi sistemleri
ile hastalıkların tes¸his ve tedavisinde büyük ilerlemeler sag˘lanmıs¸tır. Maddi ilimlerin s¸ahı olan fizig˘in, bu müktesebatı
en çok ve en iyi kullanan ilim oldug˘u herkesce bilinmekte ve kabul edilmektedir. Mühendislik dalları ise fizik ilminin
uygulamalarından ibarettir. Klasik ve modern fizig˘in yüzlerce konusundan bazıları günlük hayatın belirli sahalarında,
hayatı kolaylas¸tırmak üzere deg˘is¸ik mühendislik dallarını olus¸turmus¸tur. Makine mühendislig˘i ve ins¸aat mühendislig˘i
çıkıs¸ itibarıyla oldukça eskidirler. Bunlara nazaran elektrik mühendislig˘i daha yenidir; ondokuzuncu yüzyılın sonu ve
yirminciyüzyılınbas¸larıbumühendislig˘inserpilipgelis¸tig˘izamanlardır.
Matematig˘in yas¸antımıza olan etkisi bag˘lamında bir örnek verelim. Ondokuzuncu yüzyılda James Clark Maxwell
kendisinden önce bilinen Ampere kanununa küçük bir terim ekledi. Yer deg˘is¸tirme akımı5 adı verilen bu basit terim
sayesinde Ampere kanunu ve Faraday kanunu birbirlerine eklemlendiler. Bu iki kanunun uzay ve zamanda çözümü
sayesinde, ıs¸ıg˘ın özelliklerini tas¸ıyan elektromanyetik dalgaların varlıg˘ı öngörüldü. Ondokuzuncu yüzyılın sonlarına
dog˘ru Heinrich Herz bu dalgaların varlıg˘ını ve ıs¸ık gibi davrandıklarını deneysel olarak ispatladı. Yirminci yüzyılın
bas¸larındaJ.A.FlemingveLeeDeForesttarafındandiyotvetriyottüpleriicatedilince,MaxwellveHerz’inbulus¸larından
elektronikdiyeyenibirmühendislikalanıdog˘du.
Maxwell’inkatkıdabulundug˘uAmperekanunuilebirlikteGaussKanunlarıveFaradaykanunlarınınolus¸turdug˘udört
´
→− →−
kanunaMaxwellkanunlarıdenir. Maxwell’inküçükkatkısıolan d D·dS terimiaslındaokadarbüyükveönemlidir
dt S
kibudörtdenklemtakımıMaxwell’inismiileanılagelis¸tir.
→− →−
GaussKanunu: D·dS = Q
(cid:122)
S
→− →−
ManyetizmadaGaussKanunu: B ·dS = 0
(cid:122)
S
→− →− d →− →−
FaradayKanunu: E ·d l = − B ·dS
(cid:122) dt
C ˆ
→− →− d →− →−
AmpereKanunu: H ·d l = I+ D·dS
(cid:122) dt
S
C
Elektrik,elektronik,mikrodalgaveantenmühendisliklerinintemelibudörtkanundur. Kirchhoffakımvegerilimkanun-
ları,elektromanyetikinduksiyon,Ohmkanunu,devreelemanlarınınterminaldenklemlerivb’ninkaynag˘ıMaxwellden-
klemleridir. Mühendislikmüfredatınınilkyıllarındaverilenfizikvematematikderslerininönemlibirkısmıbudenklem-
lerin anlas¸ılmasının alt yapısını olus¸turur. Kalkulus ve lineer cebir bu meyanda sayılabilecek önemli derslerdir. Bunun
1Rahmansuresi:5
2En’amSuresi:96
3[Theuniverse]cannotbereaduntilwehavelearntthelanguageandbecomefamiliarwiththecharactersinwhichitiswritten. Itiswrittenin
mathematicallanguage,andthelettersaretriangles,circlesandothergeometricalfigures,withoutwhichmeansitishumanlyimpossibletocomprehend
asingleword.OpereIlSaggiatorep.171
4Bakarasuresi:255
5Displacementcurrent
3
yanında,sistemlervesistemlerinis¸ledikleriis¸aretkavramlarıortayaçıkmıs¸tır.Sistemlerveis¸aretler,Laplacedönüs¸ümleri,
Fourierdönüs¸ümleri,Fourierserilerigibimatematikselaletlerlekolaycaifadeedilebilmektedir. Budamühendislikanaliz
vesentezinekolaylıksag˘lamaktadır. 80’liyıllarınsonundanbas¸layarakrakamsalteknolojininucuzlamasıveyayılmasıile
z-dönüs¸ümleridesistemlerveis¸aretleralanındakendinesag˘lambiryeredinmis¸tir. Devrekonularıkendidersleriiçinde
elealınmaklaberaber,sistemlerveis¸aretlerdevredersleriiçindeverilememektedir.
Kitap esas olarak Laplace, Fourier, z-dönüs¸ümlerini ve Fourier serilerini hedef almıs¸tır. Devre analiz ve sentezi,
geribesleme(feedback)teorisi,filtreteorisi,is¸aretlervesistemler,sayısalis¸aretis¸lemevekontrolteorisielektrik-elektonik
vebiyomedikalmühendisliklerininenönemliyapıtas¸larıarasındadır.Adıgeçenkonularbuyapıtas¸larınınharcımesabesindedir.
Bukonularıkavrayabilmeninöns¸artıisekarmas¸ıksayılarvekarmas¸ıksayıfonksiyonlarıileilgilikonseptleriniyiög˘re-
nilmesidir. Bu yüzden kitap 1.bölümde kompleks sayıları ve 2.bölümde kompleks fonksiyonları ve analitisiteyi ele
almıs¸tır. ˙Ister analog, ister sayısal olsun, sistem tasarımında karmas¸ık sayı ve analizinin büyük bir ag˘ırlıg˘ı vardır. Kar-
mas¸ık sayılar ve analitik fonksiyonlar kalkulusun bıraktıg˘ı noktadan bir adım sonrasına kapı aralamaktadır. Bu sebe-
ple kalkulus bu dersin olmazsa olmaz ön s¸artıdır; bir çok konu ve kavram kalkulus bilgileri üzerine ins¸a edilmektedir.
Amacımız,ög˘renciyielektrikmühendislig˘indeçokönemlibiraraçolandönüs¸ümtekniklerimatematig˘iiletanıs¸tırmaktır.
Okuyucunun kalkulus ve diferansiyel denklemler derslerini aldıklarını kabul ediyoruz. Örneklerde zaman zaman elek-
trik devreleri kullanılmakta ise de burada gaye elektrik devrelerini ög˘retmek deg˘il, elektrik devrelerini vasıta olarak ve
motivasyongayesiylekullanmaktır. Elektrikdevreleriyanındamekanikvebilgisayaruygulamalarınadayerverilmis¸dir.
1970’liyıllarakadarmatematik, logaritmavetrigonometritablolarıyla, ellehesaplayarak; hesapmakineleriyleveya
sürgülühesapcetvelleriyleyapılıyordu.Mainframetabiredilenbilgisayarlars¸imdikibilgisayarlarımızlakars¸ılas¸tırıldıg˘ında
hemçokpahalı,hemçokyavas¸,bellekkapasitesiçokdüs¸ük,görselarayüzleriilkel;çoközelveelitkullanıcılarındıs¸ında
ulas¸ılması çok zor makinelerdi. Üniversitede ikinci sınıf ög˘rencisi oldug˘um yıl bir aylık harçlıg˘ımı ARISTO multilog
sürgülü hesap cetveline yatırdıg˘ımı hatırlıyorum. Yaz tatilinde Almanya’ya giden bir arkadas¸ıma bu parayı verip heye-
canlasürgülühesapcetvelimingelmesinibeklemis¸tim. Dahasonrabucetveli,ög˘lesıcag˘ındaarabadaunutup,rakamların
boyalarınıkabarmıs¸ birs¸ekildebuluncabüyükbirhayalkırıklıg˘ıyas¸amıs¸tım. S¸imdiartıkhesapmakinelerinivesürgülü
hesap cetvellerini bilen ve hatırlayan kaç kis¸i kalmıs¸tır bilemem6... Bugün bunların yerine bilgisayarlarımız var; çok
kuvvetli matematik yazılımlarımız var. Siz sürgülü hesap cetveli yerine MATLAB, SCILAB, MATHCAD, MATHE-
MATICA,MAXIMAvb. kullanabilirsiniz. Gördüg˘ünüzgibibizimFACITveARISTO’muzakars¸ısizinnekadarfazla
seçeneg˘inizvar! FSMVÜög˘rencilerineLabVIEWkullanmaimkanıvermektedir. Konuyucaziphalegetirmekiçinkonu
anlatımlarındaveörneklerdeLabVIEWyazılımınısıkçakullandık. Bizimeskidenhesapmakinelerimizvesürgülühesap
cetvellerimizdenaldıg˘ımızhazzınçoküstündebirzevkalacag˘ınızıgarantiedebilirim. ElbettekiLabVIEWtekseçenek
deg˘il; biz lisanslı kullanıcısı oldug˘umuz için LabVIEW üzerine yog˘unlas¸tık. Lisans dıs¸ında, LabVIEW programlama,
veritoplamaveanalizetme,sayısalsistemvecihazkontroletmegibipekçokyeteneklerisebebiyleög˘rencyebüyükkatkı
sag˘layanbirplatformdur. ÖrneklerimizdevebölümsonuproblemlerindeLabVIEWkullanmamızınsaiklerindenbiriside
budur.
Kitabı˙Ingilizceyazmayakararverdik. Bununbirkaçsebebivar.
1. FSMVÜ’nde%30˙Ingilizceeg˘itimuygulanmaktadır. Kitabın˙Ingilizcete’lifibueg˘itimedesteknitelig˘indedir.
2. Üniversitelerimizinteknikbölümlerininmezunlarındaciddibir˙Ingilizcezafiyetisözkonusudur.
3. Kitaba konu olan sahalarda Türkiye’deki akademiya içinde bir fikir birlig˘i bulunmamaktadır. Aynı kavram veya
terminoloji farklı üniversitelerde, farklı akademisyenler tarafından farklı kelime ve terminoloji ile ifade edilmeye
çalıs¸ılmaktadır. BazıterminolojininTürkçe’dehenüzstabilolmaması;yazarınmevcutterminolojiilezamanzaman
sıkıntıyas¸amasısebebiile;bunlarınzamankaybınayolaçacag˘ıendis¸esidog˘mus¸tur. Dilkonusununmillibirprob-
lemoldug˘unudüs¸ünüyorveTürkiyeçapındabütününiversitelerinveTürkDilKurumu’nunkatkılarıylaçözülmesi
vefikirbirlig˘isag˘lanmasıgerektig˘ineinanıyoruz.
4. ˙Ileride bu kitabı belki iki dilli olarak basma ihtimali. Bu tabii ki müstakbel bir proje olabilir. Türkçe terminoloji
problemi halledilditen sonra bu adım düs¸ünülebilir. Bas¸tan is¸i bu boyutu ile ele almak ise bu kitaba belki hiç
bas¸layamamak anlamına gelecekti. "Tamamına ulas¸ılmayan tamamen terkedilmez" kaidesince ˙Ingilizce bas¸layıp
sonrains¸aallahkitabınikidilliversiyonunageçerizdüs¸üncesiag˘ırlıkkazandı.
Derslerimizin Türkçe olması, kitaptaki konuların sınıfta Türkçe anlatılması demektir. Kitabın ˙Ingilizce, dersin Türkçe
olmasısebebiyleög˘rencilerimizinteknikliteratüreveterminolojiyedahakolayas¸inaolmalarınıumuyoruz.
OrhanÖzhan
Istanbul,2016
6"ARISTOsliderule","FACITcalculator"anahtarkelimeleriilebirinternettaramasıyapıp,veyahttp://www.sliderulemuseum.com/Aristo.
htmsitesindenozamankullanılanteknolojihakkındayeterlibilgiyeulas¸abilirsiniz.
4
PREFACE
InthenameofAllah,theBeneficent,theMerciful
Mathematics... Undoubtedly there is much said about it. The most beautiful of what is said has been said in the most
beautiful of the words: "The sun and the moon [move] by precise calculation7", "[He is] the cleaver of daybreak and
hasmadethenightforrestandthesunandmoonforcalculation. ThatisthedeterminationoftheExaltedinMight,the
Knowing8". Inthephysicaluniverseeverynaturaleventwhichisthesubjectmatterofsciencecanbeexpressedinmath-
ematics. Everymaterialcreationintheuniversecanbeunderstoodinthislanguage. AsGalileooncesaid: "Philosophy
[i.e. physics] is written in this grand book — I mean the universe — which stands continually open to our gaze, but it
cannotbeunderstoodunlessonefirstlearnstocomprehendthelanguageandinterpretthecharactersinwhichitiswrit-
ten. Itiswritteninthelanguageofmathematics,anditscharactersaretriangles,circles,andothergeometricalfigures,
without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a
darklabyrinth.9"ContrarytotheclaimthatmathematicsiscreatedbytheMan,mathematics,thelanguageofthecreated,
asopposedtothecreation,hasnobeginningasmathematicsiswithinAllah’sknowledge;AllahhasnobeginningandHis
Knowledgehasnobeginningeither. Assuchmathematicsimpartsmanadifferentpleasureandsatisfaction;itstasteand
beautybeingperceivedbyintellect. Inaccordancewiththedivineword"AndtheyencompassnothingofHisknowledge
exceptforwhatHewills10",ifweeverknowmathematicsitissolelyduetoAllah’sbenefactionandendowment;andthis
is a grand honor and blessing for man. Even if the legacy of mathematics acquired during the history of humanity and
scienceshouldfilllibraries,thisacquisitionislikenothingbutadropletfromtheoceanorless.
Itiswiththisminusculelegacythattheatomicstructureandtheelementaryparticleshavebeenunderstood;astronomy
hasreachedalevelwhichmadeitpossibletosendspacecraftbeyondsolarsystemandlandprobesontheMartiansoil;
electronics and communication technology have devised smart phones which let people talk to each other face-to-face
fromadistance; theworldhasbeentransformedtoasmallvillagebyairandseatransportation; anexplosionhasbeen
witnessedininformationtechnologybecauseofdevelopmentsincomputerandinternettechnologies;incredibleimaging
and treatment systems in medicine have ushered in unprecedented advances in diagnosing and treating diseases. It is
known and accepted by everyone that physics, the master of this acquisition, is the branch of knowledge that best uses
this legacy. Various engineering branches are but applications of physics. Some fields of classical and modern physics
havecrystallizedintospecificengineeringdisciplinestoeasecertainaspectsofourdailylives. Mechanicalengineering
andcivilengineeringarerelativelyolderthanelectricalengineeringbybirth;thelate19thcenturyandearly20thcentury
arethetimeslotswhenthisengineeringwasbornandflourished.
Letusmentionanexamplefortheinfluenceofmathematicsonourlives. In19thcenturyJamesClarkMaxwelladded
asmalltermtoAmpere’sLawalreadyknownbeforehim. Thankstothissimpleterm,theso-calleddisplacementcurrent,
Ampere’sLawandFaraday’slawbecamecoupled. Simultaneousspatialandtemporalsolutionoftheseequationsledto
prediction of electromagnetic waves which possessed properties light. In fact toward the end of 19th century Heinrich
Herzdemonstratedtheexistenceofthesewavesthroughexperimentsandthattheybehavedlikelight. Inearly20thcen-
turyJ.A.FlemingandLeeDeForestinventeddiodeandtriodevacuumtubeswherebyanewengineeringwasborn;the
newengineeringleveragedMaxwell’sandHerz’sideasandwascalledelectronics11.
Fourcanoniclawsi.e.,Gauss’sLaws,Faraday’sLawandAmpere’sLaw,arecollectivelycalledMaxwell’sequations.
´
→− →−
Maxwell’sminusculecontributiontoAmpere’sLaw, thedisplacementcurrent d D·dS issuchabigdiscoverythat
dt S
thesefourlawsareknownbyMaxwell’sname. Withoutthistermelectromagneticwaveswouldnotbediscovered.
→− →−
Gauss’sLaw(electric): D·dS = Q
(cid:122)
S
→− →−
Gauss’sLaw(magnetic): B ·dS = 0
(cid:122)
S
→− →− d →− →−
Faraday’sLaw: E ·d l = − B ·dS
(cid:122) dt
C ˆ
→− →− d →− →−
Ampere’sLaw: H ·d l = I+ D·dS
(cid:122) dt
S
C
7SurahAr-Rahman:5
8SurahAl-An’am:96
9TheAssayer,IlSaggiatorep.171,GalileoGalilei,Rome,1623.
10Surahal-Baqara:255
11Thenaming,electronics,wasprobablybecausethediode,triodeandlaterpentodewereelectrondevices.
5
These four laws are fundamental to electrical, electronics, microwave and antenna engineering. Kirchhoff’s cur-
rent and voltage laws, electromagnetic induction, Ohm’s law, terminal equations of circuit elements are derived from
Maxwell’sequations. Physicsandmathematicscoursesinfreshmanandsophomoreofelectricalengineeringcurriculum
serve as the infrastructure towards understanding these equations. Calculus and linear algebra are important subjects
towardstothisend. Besidesthese,signalsandsystemsconceptshaveemergedinelectricalengineering. Signalsandsys-
tems,canbestudiedusingmathematicalsuchtoolsasLaplacetransforms,FouriertransformsandFourierseries. These
toolsprovideeaseofanalysisandsynthesiswithsignalsandsystems. Withthespreadandaffordabilityofdigitaltech-
nology,startinginlate1980’s,z-transformtoohasfoundanicheandsecureditsplaceinsignalsandsystemsapplications.
AtFSMVUwehaveobservedthattheexistingmathcoursesinthecurriculumareinsufficienttoreadythestudentfor
electricalengineeringcourses. Assuchithasbeendeemedappropriatetosetupacoursewhichwouldmakeupthisvery
gapandfillit. AsaresultMAT235hasbeenintroducedinbiomedicalengineeringandelectrical-electronicsengineering
curricula. Thisbookisacompilationoftheauthor’slecturenotessincethecoursewasassignedtohim. Theideawasthat
notesinpdforprintedformwouldbebetterthanloosenotes.
The main objective of the book is to teach Fourier series, Laplace, Fourier, and z-transforms. Circuit analysis and
synthesis,feedbacktheory,filtertheory,signalsandsystemsandcontroltheoryarethemostimportantbuildingblocksof
electrical-electronicsandbiomedicalengineering.Thetargetedtopicsinthebookareindispensabletoolsinordertostudy
thesefields. Thesetoolscannotbegraspedenoughwithoutaworkingknowledgeofcomplexnumbertheory. Therefore
thebookstartsinChapter1withcomplexnumbersandcontinueswithfunctionsofcomplexnumbersandanalyticityin
Chapter 2. Whether analog or digital, complex numbers and functions play a crucial role in system design. Complex
numbersandanalyticfunctionsopenadoortoawonderfulworldwherethecalculushasleftoff. Thereforecalculusis
anessentialprerequisiteforthiscourseasmanynovelconceptsarebuiltoncalculusandsome"old"ideasarerefinedor
generalized. Our aim is to introduce the student to transform mathematics. We assume that they have taken courses in
calculusanddifferentialequations. Althoughweliberallyresorttoelectricalcircuitsinexamplestimeandagain,itisnot
ourintentioninthiscoursetoteachelectriccircuits,butweusethemasavehicleandmotivation.Besideselectriccircuits,
mechanicalandcomputingexamplesalsotakeplaceinthebook.
Until1970’smathematicalcalculationsweredonemanuallyusinglogarithmicandtrigonometrictablesorusingme-
chanicalcalculators,orsliderules. Mainframecomputersweremachineswhich,whencomparedtoourmoderndesktop
or laptop computers, were too expensive, too slow, too bulky; had too small a memory, had a primitive user interface,
andwerehardlyaccessibletoaverageusers. IrememberhavingsavedamonthofmypocketmoneytoorderanARISTO
Multilog slide rule through a friend who was traveling to Germany in that summer vacation. I had eagerly waited for
the arrival of the slide rule with excitement and impatience. Later on I had been utterly disappointed and shocked to
find the dyes of the letters and numbers had come off due to heat because I had forgotten it inside a car under intense
noontemperatures. NowIdon’tknowhowmanypeoplehaveremainedwhoknowaboutandrememberthemechanical
calculatorsandsliderules12... Nowweenjoycomputersonwhichwerunextremelypowerfulmathematicssoftware. In
lieuofslideruleyoucanuseMATLAB,SCILAB,MATHCAD,MATHEMATICA,MAXIMAetc. Allofthesesoftware
have provisions for complex number operations and complex number functions. As you will agree and appreciate you
havemoremathtoolalternativesthanweoncehad! LabVIEWisjustanotherfantasticmathtoolforstudents. Inorder
tomakethesubjectmoreinterestingandattractive,wehaveoftenusedLabVIEWintextandexamples. WithLabVIEW
orsimilartoolsyoucanderivepleasureandsatisfactionmuchsuperiortoourswehadderivedusingthearchaictools; I
can guarantee it! Since we are a licenced user, we have focused on LabVIEW with a few examples with MATLAB or
SCILAB. A data acquisition, analysis, and instrument control platform, LabVIEW contributes to student’s intellectual
development. TheseveryattributesaremotivesinourchoiceofLabVIEWinthetext,examplesandchapterproblems.
Howeverawarningisinorderforreaderofwit. Allthesesupersoftwarewilldoyounogoodunlessyouknowand
understandtheunderlyingmathematics. Theyjustmakeyourjobeasier,funnierandmoreenjoyable. Convinceyourself
that,withoutproperunderstanding,evenwiththesesoftwareyoucan’tgoveryfar.Don’tsettletoatrial-and-errorstrategy
withsoftwarefortheknowledgeitselfinspiresandgivesyouinsight. Neversuccumbtothecommonfallacythatyouare
taughtuselesstheoriesinthisandothercoursesthatyouhavetotake. Hopefullythesoftwareofourchoicewillalsohelp
youlikemathematicsasis,andmakeitajoytolearnitthroughitsnumber-crunchingpowerandbreath-takinggraphics.
Finallyasforthelanguage,wehavedecidedtowritethebookinEnglishforafewreasons.
1. SomeTurkishuniversitiesteachinEnglishexclusively; someteach30%oftheircurriculainEnglish. Ourchoice
ofEnglishhopefullypromotesthiseducation.
12Enterthekeywords"ARISTOsliderule","FACITcalculator"ortheurlhttp://www.sliderulemuseum.com/Aristo.htmintoawebbrowser
tolearnaboutthecomputingtechnologyusedthen.
6
2. AseriousweaknessandinsufficiencyofuniversitygraduatesinEnglishareofconcern. AtextwritteninEnglish
whilethecourseistaughtinTurkishshouldhelpthestudenttolearnprofessionintwolanguages.
3. Regarding the fields touched upon in the book, there exists no consensus on Turkish terminology among Turkish
academia. Different Turkish terminology can be used for the same concepts at different universities, by different
academicians. SometerminologybeingyetunstableinTurkish,havecausedtheauthortroubleinlecturesattimes;
assuchinevitabledelaysinwritingcouldbeincurred. Wethinkthatthelanguageisanationalproblemwhichhas
tobesolvedinordertoarriveataconsensuswithcollectivecontributionsfromuniversitiesacrossTurkey,andthe
TurkishLanguageInstitute.
4. Viable probability of publishing the book in bilingual form. Certainly this can be a future project which can be
takenupaftersolvingtheterminologyproblem. Tostartthebilingualprojectfromthebeginningwouldrenderthe
workamissionimpossible. Incompliancewiththemaxim"Whatcan’tbegraspedinitsentiretyisnotabandoned
entirely",thethoughtofstartinginEnglishfirst,thencontinuinginbilingualformhasgainedweight.
Definitely,chaptersoneandtwomustbetaughtinsequenceandbeforetheotherchapters.FourierSeriesandFourier
TransformcanbetaughtinanysequenceasdesiredbeforeoraftertheLaplaceTransform. Inaonesemestercourse
z-Transformcanperhapsbeleftasthelasttopic.
OrhanÖzhan
Istanbul,2016
Contents
1 ComplexNumbers 9
1.1 RepresentationofComplexNumbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Euler’sFormula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 MathematicalOperations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.1 Conjugate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.2 Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.3 AdditionandSubtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.4 MultiplicationandDivision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.5 Rotatinganumberincomplexplane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 Rootsofacomplexnumber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5 ApplicationsofComplexNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5.1 ComplexNumbersversusTrigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5.2 Phasors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5.3 3-PhaseElectricalCircuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.5.4 NegativeFrequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.5.5 ComplexNumbersinMathematicsSoftware . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 FunctionsofaComplexVariable 33
2.1 LimitofaComplexFunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2 DerivativeofComplexFunctionsandAnalyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3 HarmonicFunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.4 SomeElementaryFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4.1 PolynomialandRationalFunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.4.2 ExponentialFunctionofaComplexVariable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.4.3 LogarithmofaComplexNumber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.4.4 TrigonometricFunctionsofaComplexVariable . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.4.5 HyperbolicFunctionsofaComplexVariable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3 TheLaplaceTransform 53
3.1 MotivationtouseLaplaceTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2 DefinitionoftheLaplaceTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3 PropertiesoftheLaplaceTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.4 TheInverseLaplaceTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.4.1 Realroots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4.2 Complexroots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4.3 Multipleroots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.5 PolesandZeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.5.1 Factoringpolynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.5.2 Polesandtimeresponse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.5.3 Analternativewaytosolvedifferentialequations . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.6 ApplicationsofLaplaceTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.6.1 Electricalsystems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.6.2 Evaluationofdefiniteintegrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7
8 CONTENTS
4 TheFourierSeries 89
4.1 VectorsandSignals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.2 Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.3 CalculatingFourierSeriesCoefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.4 PropertiesofFourierSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.4.1 SymmetryConditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.4.2 TimeOperations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.5 Parseval’sRelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.5.1 ConvergenceofFourierSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.5.2 GibbsPhenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.6 ApplicationsofFourierSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5 TheFourierTransform 117
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.2 DefinitionoftheFourierTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.3 FourierTransformversusFourierSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.4 ConvergenceoftheFourierTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.5 PropertiesoftheFourierTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.5.1 SymmetryIssues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.5.2 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.5.3 TimeScaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.5.4 TimeShifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.5.5 FrequencyShifting(AmplitudeModulation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.5.6 Differentiationwithrespecttotime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.5.7 Integrationwithrespecttotime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.5.8 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.5.9 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.5.10 MultiplicationinTimeDomain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.5.11 Parseval’sRelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.6 FourierTransformversusLaplaceTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.7 FourierTransformofDiscreteSignals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.7.1 Short-TermFourierTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.7.2 TheDiscreteFourierTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.8 TwoDimensionalFourierTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.9 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.9.1 CepstrumAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.9.2 CircuitApplications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
5.9.3 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.9.4 AmplitudeModulationandDemodulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6 z-Transform 167
6.1 Definitionofthez-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.2 RegionofConvergenceforthez-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.3 z-TransformProperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.4 TheInversez-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.4.1 PartialFractionExpansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.5 Parseval’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
6.6 One-Sidedz-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
6.6.1 DifferenceEquationsinLabVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
6.7 FrequencyResponse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
6.8 Applicationsofz-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
7 PowerSeries 189
7.1 TaylorSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
7.2 MaclaurinSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
7.3 LaurentSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
Chapter 1
Complex Numbers
Invention of decimal number system, zero and negative numbers have been great
intellectualstridesforustounderstandandworkwithmathematicalproblems. The
setofallpositiveandnegativenumbers,andzeroconstitutesaframeworkthatwe
call “real numbers” and denote with R. With real numbers we can tackle a large
setofproblemsinourdailylives. Howeverasourintellectualrealmhaswidened
further, we have encountered instances which our established number system is
unabletohandle.
Therootsof-1wereshunnedbypeopleasbeingimpossible,undefinedorunre-
alistic. In16thcenturyItalianmathematicianGerolamoCardanosoughttwonum-
Eulerformula,theamazinglink bers whose sum is 10 and their product is 40. He gave two “impossible” results,
√ √
fromexponentialfunctionto 5+ −15and5− −15. Cardanowarnedthatthesewere“meaningless,fictitious
complexnumbers. Thesecondline and imaginary[Cardano].” In 17th century Swiss Mathematician Leonhard Euler
derivedfromthefirstiscalledby wroteabookonalgebraandgaveseveralexamplesandapplicationsofimaginary
Feynmanthemostelegantformula numbers. Hemadehoweverapolegeticremarksaboutthesefictitiousnumberssay-
inmathematics. ingthat:
√ √
“Allsuchexpressionsas −1, −2,etc. areimpossibleorimaginary
numbers,sincetheyrepresentrootsofnegativequantities,andofsuch
numberswemaytrulyassertthattheyareneithernothing1,norgreaterthannothing,norlessthannothing,
whichnecessarilyconstitutesthemimaginaryorimpossible[Euler].”
To generalize the concept of number, we can invent a number i (j in electrical engineering), which, when multiplied
by itself, produces -1. This very number, being strange, unreal, or whatever, has been called an imaginary number as
suggested by Euler. Our familiar old numbers have been called real numbers by the same token. By mixing real and
imaginary numbers we form a new framework which we call complex numbers. The set of all complex numbers is
denotedbyC.
Real numbers are traditionally illustrated by a line which has a point for 0. The numbers to the right of zero are
positive; those to its left are negative. Likewise we can use the plane to represent complex numbers. We draw two
perpendicularlinesforrealandimaginaryaxes.Ontherealandimaginarynumberaxes0iscommon.Onthisplaneevery
pointhastwocomponents: onerepresentingtherealpartandanotherrepresentingtheimaginarypart.
1.1 Representation of Complex Numbers
Acomplexnumberzisapairofrealnumbers(x,y)inCartesiancoordinatesystemanddenotedby
z=x+jy (1.1)
√
where j= −1. xandyarecalledtherealandimaginarypartofzrespectively.
x=Re(z)
y=Im(z)
1By“nothing”Eulermeans“zero.”
9
10 CHAPTER1. COMPLEXNUMBERS
Im
z=x+jy
y
r
0 θ Re
x
Figure1.1: Acomplexnumbercanberepresentedbyapointinthecomplexplane.
1
Fromthedefinitionof jonecandeducethat j2=−1, =−j, j3=−jand j4=1.
j
zcanalsoberepresentedas
z=r∠θ =|z|∠θ (1.2)
wherer=|z|iscalledthenormormagnitudeofz,andθ isitsargument. θ istheanglesubtendedbythelinedrawnfrom
thepoint(x,y)totheorigin. PolarcoordinatesareobtainedfromrectangularcoordinatesusingtherighttriangleinFig.
1.1:
(cid:112)
r = x2+y2
y
θ = tan−1( )
x
Eqn. 1.1andEqn. 1.2arecalledtherectangularandpolarrepresentationsofzrespectively.
1.2 Euler’s Formula
Letzbeacomplexnumberwhosemagnitudeis1,i.e.,|z|=r=(cid:112)x2+y2=1. ThenEqn. (1.2)yieldszas1∠θ. Another
beautiful notation to for z=1∠θ is ejθ which is technically more elegant. As the following development is about to
demonstrate, this representation gives us a deeper insight into the nature of complex numbers, and helps us establish
usefulresults.
ComplexExponential. WecancarryoutMaclaurinSeriesexpansionoftheexponentialfunctionejθ:
(jθ)2 (jθ)3 (jθ)4 (jθ)5 (jθ)n
ejθ = 1+jθ+ + + + +···+ +···
2! 3! 4! 5! n!
θ2 θ3 θ4 θ5
= 1+jθ− −j + +j −···
2! 3! 4! 5!
θ2 θ4 (cid:18) θ3 θ5 (cid:19)
= 1− + −···+j θ− + −···
2! 4! 3! 5!
θ2 θ4 θ3 θ5
Sincewerecognize1− + −··· ascosθ,andθ− + −··· assinθ weconcludethat
2! 4! 3! 5!
ejθ =cosθ+jsinθ (1.3)
AspecialcaseofEqn. (1.3)iscalledEuler’sIdentityandobtainedbysettingθ =π:
ejπ+1=0 (1.4)
whichRichardFeynmancalled"themostremarkableformulainmathematics[Feynman]."
1.3 Mathematical Operations
1.3.1 Conjugate
Conjugateofacomplexnumberz=x+jyisdefinedas
