Fundamentals of Complex Analysis with Applications to Engineering, Science, and Mathematics: Pearson New International Edition

521 Pages·2013·3.725 MB·English

by  Edward B. Saff, Arthur David Snider

F Fundamentals of Complex Analysis
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Edward B. Saff Arthur David Snider
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ISBN 978-1-29202-375-5
9 781292 023755
Fundamentals of Complex Analysis
Engineering, Science, and Mathematics
Edward B. Saff Arthur David Snider
Third Edition
ISBN 10: 1-292-02375-9
ISBN 13: 978-1-292-02375-5
Pearson Education Limited
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ISBN 10: 1-292-02375-9
ISBN 13: 978-1-292-02375-5
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12334559430641195795339
P E A R S O N C U S T O M L I B R AR Y
Table of Contents
Chapter 1. Complex Numbers
Edward B. Saff/Arthur David Snider 1
Chapter 2. Analytic Functions
Edward B. Saff/Arthur David Snider 53
Chapter 3. Elementary Functions
Edward B. Saff/Arthur David Snider 99
Chapter 4. Complex Integration
Edward B. Saff/Arthur David Snider 149
Chapter 5. Series Representations for Analytic Functions
Edward B. Saff/Arthur David Snider 235
Chapter 6. Residue Theory
Edward B. Saff/Arthur David Snider 307
Chapter 7. Conformal Mapping
Edward B. Saff/Arthur David Snider 369
Chapter 8. The Transforms of Applied Mathematics
Edward B. Saff/Arthur David Snider 445
Index 513
I
II
Chapter 1
Complex Numbers
1.1 The Algebra of Complex Numbers
To achieve a proper perspective for studying the system of complex numbers, let us
begin by briefly reviewing the construction of the various numbers used in computa-
tion.
We start with the rational numbers. These are ratios of integers and are written
in the form m/n, n (cid:1)= 0, with the stipulation that all rationals of the form n/n are
equalto1(sowecancancelcommonfactors). Thearithmeticoperationsofaddition,
subtraction,multiplication,anddivisionwiththesenumberscanalwaysbeperformed
inafinitenumberofsteps,andtheresultsare,again,rationalnumbers. Furthermore,
there are certain simple rules concerning the order in which the computations can
proceed. Thesearethefamiliarcommutative,associative,anddistributivelaws:
CommutativeLawofAddition
a+b =b+a
CommutativeLawofMultiplication
ab =ba
AssociativeLawofAddition
a+(b+c)=(a+b)+c
AssociativeLawofMultiplication
a(bc)=(ab)c
DistributiveLaw
(a+b)c =ac+bc,
foranyrationalsa,b,andc.
From Chapter 1 of Fundamentals of Complex Analysis with Applications to Engineering, Science, and Mathematics,
Third Edition. Edward B. Saff, Arthur David Snider. Copyright © 2003 by Pearson Education, Inc. All rights reserved.
1
2 ComplexNumbers
Noticethattherationalsaretheonlynumberswewouldeverneed,tosolveequa-
tionsoftheform
ax +b =0.
Thesolution,fornonzeroa,is x=−b/a,andsincethisistheratiooftworationals,it
isitselfrational.
However,ifwetrytosolvequadraticequationsintherationalsystem,wefindthat
someofthemhavenosolution;forexample,thesimpleequation
x2 =2 (1)
cannot be satisfied by any rational number (see Prob. 29 at the end of this section).
Therefore,togetamoresatisfactorynumbersystem,weextendtheconcep√tof“num-
ber”byappendingtotherationalsanewsymbol,mnemonicallywrittenas 2,which
is defined to be a solution of Eq. (1)). Our revised concept of a number is now an
expressioninthestandardform √
a+b 2, (2)
wherea andbarerationals. Additionandsubtractionareperformedaccordingto
√ √ √
(a+b 2)±(c+d 2)=(a±c)+(b±d) 2. (3)
Multipli√cationisdefinedviathedistributivelawwiththeprovisothatthesquareofthe
symbol 2canalwaysbereplacedbytherationalnumber2. Thuswehave
√ √ √
(a+b 2)(c+d 2)=(ac+2bd)+(bc+ad) 2. (4)
Finally, using the well-known process of rationalizing the denominator, we can put
thequotientofanytwoofthesenewnumbersintothestandardform
√ √ √
a+b 2 a+b 2 c−d 2 ac−2bd bc−ad √
√ = √ √ = + 2. (5)
c+d 2 c+d 2 c−d 2 c2−2d2 c2−2d2
Thisprocedureof“calculatingwithradicals”shouldbeveryfamiliartothereader,
and the resulting arithmetic system can easily be shown to satisfy th√e commutative,
associative,anddistributivelaws. However,observethatthesymbol 2hasnotbeen
absorbed by the rational numbers painlessly. Indeed, in the standard form (2) and in
thealgorithms(3),(4),and√(5)itspresencestandsoutlikeasorethumb. Actually,we
areonlyusingthesymbol 2to“holdaplace”whilewecomputearounditusingthe
rationalcomponents,exceptforthoseoccasionalopportunitieswhenitoccurs√squared
and we are temporarily relieved of having to carry it. So the inclusion of 2 as a
numberisasomewhatartificialprocess,devisedsolelysothatwemighthavearicher
systeminwhichwecansolvetheequationx2 =2.
With this in mind, let us jump to the stage√where we have appended all the real
numberstooursystem. Someofthem,suchas 4 17,ariseassolutionsofmorecom-
plicated equations, while others, such as π and e, come from certain limit processes.
2
1.1 TheAlgebraofComplexNumbers 3
Each irrational is absorbed in a somewhat artificial manner, but once again the re-
sultingconglomerateofnumbersandarithmeticoperationssatisfiesthecommutative,
associative,anddistributivelaws.†
Atthispointweobservethatwestillcannotsolvetheequation
x2 =−1. (6)
But now our experience suggests that we can expand our nu√mber system once again
by appending a symbol for a solution to Eq. (6); instead of −1, it is customary to
use the symbol i. (Engineers often use th√e letter j.) Next we imitate the model of
expressions (2) through (5) (pertaining to 2) and thereby generalize our concept of
numberasfollows:‡
Definition1. Acomplexnumberisanexpressionoftheforma+bi,wherea
andbarerealnumbers. Twocomplexnumbersa+bi andc+di aresaidtobe
equal(a+bi =c+di)ifandonlyifa =candb =d.
Theoperationsofadditionandsubtractionofcomplexnumbersaregivenby
(a+bi)±(c+di):=(a±c)+(b±d)i,
wherethesymbol:=means“isdefinedtobe.”
Inaccordancewiththedistributivelawandtheprovisothati2 =−1,wepostulate
thefollowing:
Themultiplicationoftwocomplexnumbersisdefinedby
(a+bi)(c+di):=(ac−bd)+(bc+ad)i.
To compute the quotient of two complex numbers, we again “rationalize the de-
nominator”:
a+bi a+bi c−di ac+bd bc−ad
= = + i.
c+di c+di c−di c2+d2 c2+d2
Thusweformallypostulatethefollowing:
Thedivisionofcomplexnumbersisgivenby
a+bi ac+bd bc−ad
:= + i (ifc2+d2 (cid:1)=0).
c+di c2+d2 c2+d2
These are rules for computing in the complex number system. The usual alge-
braic properties (commutativity, associativity, etc.) are easy to verify and appear as
exercises.
†The algebraic aspects of extending a number field are discussed in Ref. 5 at the end of this
chapter.
‡KarlFriedrichGauss(1777–1855)wasthefirstmathematiciantousecomplexnumbersfreely
andgivethemfullacceptanceasgenuinemathematicalobjects.
3
4 ComplexNumbers
Example1
Findthequotient
(6+2i)−(1+3i)
.
(−1+i)−2
Solution.
(6+2i)−(1+3i) 5−i (5−i) (−3−i)
= =
(−1+i)−2 −3+i (−3+i)(−3−i)
−15−1−5i +3i
= (7)
9+1
8 1
=− − i. (cid:1)
5 5
(Aslugmarkstheendofsolutionsorproofsthroughoutthetext.)
Historically, i was considered as an “imaginary” number because of the blatant
impossibilityofsolvingEq.(6)withanyofthenumbersathand. Withtheperspect√ive
we√havedeveloped,wecanseethatthislabelcouldalsobeappliedtothenumbers 2
or 4 17; likethem,i issimplyonemoresymbolappendedtoagivennumbersystem
tocreatearichersystem. Nonetheless,traditiondictatesthefollowingdesignations:†
Definition2. Therealpartofthecomplexnumbera+bi isthe(real)number
a; itsimaginarypartisthe(real)numberb. Ifa iszero, thenumberissaidto
beapureimaginarynumber.
Forconveniencewecustomarilyuseasingleletter,usuallyz,todenoteacomplex
number. ItsrealandimaginarypartsarethenwrittenRezandImz,respectively. With
thisnotationwehavez =Rez+i Imz.
Observethattheequationz = z holdsifandonlyifRez = Rez andImz =
1 2 1 2 1
Imz . Thus any equation involving complex numbers can be interpreted as a pair of
2
realequations.
ThesetofallcomplexnumbersissometimesdenotedasC.Unliketherealnum-
ber system, there is no natural ordering for the elements of C; it is meaningless, for
example,toaskwhether2+3i isgreaterthanorlessthan3+2i. (SeeProb.30.)
EXERCISES 1.1
1. Verifythat−i isalsoarootofEq.(6).
2. Verifythecommutative,associative,anddistributivelawsforcomplexnumbers.
†Rene´ Descartes introduced the terminology “real” and “imaginary” in 1637. W. R. Hamilton
referredtoanumber’s“imaginarypart”in1843.
4
1.1 TheAlgebraofComplexNumbers 5
3. Notice that 0 and 1 retain their “identity” properties as complex numbers; that is,
0+z =zand1·z =zwhenziscomplex.
(a) Verify that complex subtraction is the inverse of complex addition (that is,
z =z −z ifandonlyifz +z =z ).
3 2 1 3 1 2
(b) Verify that complex division, as given in the text, is the inverse of complex
multiplication(thatis,ifz (cid:1)=0,thenz =z /z ifandonlyifz z =z ).
2 3 1 2 3 2 1
4. Provethatifz z =0,thenz =0orz =0.
1 2 1 2
InProblems5–13,writethenumberintheforma+bi.
(cid:1) (cid:2)
i 2
5. (a)−3 (b)(8+i)−(5+i) (c)
2 i
2−i
6. (a)(−1+i)2 (b) (c)i(π −4i)
1
3
8i −1 −1+5i 3 i
7. (a) (b) (c) +
i 2+3i i 3
(8+2i)−(1−i)
8.
(2+i)2
2+3i 8+i
9. −
1+2i 6−i
(cid:3) (cid:4)
2+i 2
10.
6i −(1−2i)
11. i3(i +1)2
12. (2+i)(−1−i)(3−2i)
13. ((3−i)2−3)i
14. ShowthatRe(iz)=−Imzforeverycomplexnumberz.
15. Letk beaninteger. Showthat
i4k =1, i4k+1 =i, i4k+2 =−1, i4k+3 =−i.
16. UsetheresultofProblem15tofind
(a)i7 (b)i62 (c)i−202 (d)i−4321
17. UsetheresultofProblem15toevaluate
8
3i11+6i3+ +i−1.
i20
18. Showthatthecomplexnumberz =−1+i satisfiestheequation
z2+2z+2=0.
5