The Calculus of Complex Functions

456 Pages·2022·22.525 MB·English

by  William Johnston

AMS / MAA TEXTBOOKS VOL 71
The Calculus of Complex Functions
William Johnston
I.Primitives–DerivativesoftheIntegralsaretheIntegrands
𝑧𝑛+1 1
1. ∫𝑧𝑛𝑑𝑧= 2. ∫ 𝑑𝑧=Log𝑧
𝑛+1 𝑧
3. ∫e𝑧𝑑𝑧=e𝑧 4. ∫sin𝑧𝑑𝑧=−cos𝑧
5. ∫cos𝑧𝑑𝑧=sin𝑧 6. ∫tan𝑧𝑑𝑧=Logsec𝑧
7. ∫cot𝑧𝑑𝑧=Logsin𝑧 8. ∫sec𝑧𝑑𝑧=Log(sec𝑧+tan𝑧)
9. ∫csc𝑧𝑑𝑧=Log(csc𝑧−cot𝑧) 10. ∫sec2𝑧𝑑𝑧=tan𝑧
11. ∫csc2𝑧𝑑𝑧=−cot𝑧 12. ∫sec𝑧tan𝑧𝑑𝑧=sec𝑧
13. ∫sinh𝑧𝑑𝑧=cosh𝑧 14. ∫cosh𝑧𝑑𝑧=sinh𝑧
𝑑𝑧 1 𝑧
15. ∫tanh𝑧𝑑𝑧=Logcosh𝑧 16. ∫ = arctan
𝑧2+𝑎2 𝑎 𝑎
𝑑𝑧 1 𝑧−𝑎 𝑑𝑧
17. ∫ = Log( ) 18. ∫ =Log(𝑧+√𝑧2±𝑎2)
𝑧2−𝑎2 2𝑎 𝑧+𝑎 √𝑧2±𝑎2
e𝑧(sin𝑧−cos𝑧) e𝑧(sin𝑧+cos𝑧)
19. ∫e𝑧sin𝑧𝑑𝑧= 20. ∫e𝑧cos𝑧𝑑𝑧=
2 2
𝑧 𝑎 𝑧
21.∫√𝑎2−𝑧2𝑑𝑧= √𝑎2−𝑧2+ sin−1( )
2 2 𝑎
𝑧
22.∫√𝑧2±𝑎2𝑑𝑧= √𝑧2±𝑎2±Log(𝑧+√𝑧2±𝑎2)
2
Anarbitraryconstantmaybeaddedtoanyofthetablevalues.
II.CommonIntegrationTechniques
(A) 𝑢-Substitution: ∫𝑓(𝑔(𝑧))𝑔′(𝑧)𝑑𝑧=∫𝑓(𝑢)𝑑𝑢,where𝑢=𝑔(𝑧).
(B) IntegrationbyParts: ∫𝑢𝑑𝑣=𝑢𝑣−∫𝑣𝑑𝑢.
𝑃(𝑥)
(C) Partial Fraction Decomposition: To integrate a rational function , follow a three
𝑄(𝑥)
stepprocess.First,uselongdivision(ifnecessary)tomakethenumeratorpolynomial’s
degreelessthanthedenominator’s.Second, factorthedenominatorintolinearterms.
Third,equatetherationalfunctiontothesumofaconstantovereachlineartermand
solvefortheconstants.Theprimitiveoftherationalfunctionisthesumoftheprimitives
oftheseresultingterms.
(D) Half-angleSubstitution: Whensimplemethodswillnotintegratearatioofsumsand/or
productsoftrigonometricfunctions,thistechniquecanalwayswork.Writethefunction
asaratioofsumsand/orproductsofsinesandcosines,andthenusethesubstitution
𝑥 2𝑡 1−𝑡2
𝑡 = tan( ).Applydouble-angletrigonometricidentitiessin𝑥 = ,cos𝑥 = ,
2 1+𝑡2 1+𝑡2
2𝑑𝑡
and𝑑𝑥= .Thetrigonometricintegralbecomestheintegralofarationalfunction,
1+𝑡2
whoseprimitivemaythenbefound,forexample,viapartialfractiondecomposition.
III.FourierCoefficients
∞ 𝜋
1
ThecomplexFourierseries𝑓(𝑥)∼ ∑ 𝑐 e𝑖𝑛𝑥has𝑐 = ∫ 𝑓(𝑡)e−𝑖𝑛𝑡𝑑𝑡.
𝑛 𝑛 2𝜋
𝑛=−∞ −𝜋
∞
TherealFourierseries𝑓(𝑥)∼𝑎 + ∑ (𝑎 cos𝑛𝑥+𝑏 sin𝑛𝑥)hascoefficients
0 𝑛 𝑛
𝑛=1
𝜋 𝜋 𝜋
1 1 1
𝑎 = ∫ 𝑓(𝑡)𝑑𝑡, 𝑎 = ∫ 𝑓(𝑡)cos(𝑛𝑡)𝑑𝑥, and 𝑏 = ∫ 𝑓(𝑥)sin(𝑛𝑡)𝑑𝑡.
0 2𝜋 𝑛 𝜋 𝑛 𝜋
−𝜋 −𝜋 −𝜋
𝑎 −𝑖𝑏 𝑎 +𝑖𝑏
Theyarerelatedvia𝑐 =𝑎 ,𝑐 = 𝑛 𝑛,and𝑐 = 𝑛 𝑛 for𝑛=1,2,3,...
0 0 𝑛 2 −𝑛 2
IV.Trigonometry
1 1
A.HalfAngleFormulas: cos2𝜃 = (1+cos2𝜃), sin2𝜃 = (1−cos2𝜃)
2 2
B.AdditionFormulas: sin(𝛼+𝛽)=sin𝛼cos𝛽+cos𝛼sin𝛽,
cos(𝛼+𝛽)=cos𝛼cos𝛽−sin𝛼sin𝛽
C.UnitCircleValuesintheComplexPlane:
√2 √2 𝑖𝜋
Eachpoint𝑥+𝑖𝑦onthecircleequalse𝑖𝜃withpolarangle𝜃.Forexample, + 𝑖 =e 4
2 2

The Calculus
of Complex Functions

AMS/MAA TEXTBOOKS
VOL 71
The Calculus
of Complex Functions
William Johnston
MAATextbooksEditorialBoard
WilliamR.Green,Co-Editor
SuzanneLynneLarson,Co-Editor
PaulT.Allen MarkBollman DebraS.Carney
HughN.Howards WilliamJohnston EmekKose
MichaelJ.McAsey ThomasC.Ratliff PamelaRichardson
JeffreyL.Stuart RonTaylor RuthVanderpool
ElizabethWilcox
2020MathematicsSubjectClassification.Primary30-01;
Secondary30-03,30B10,30C35,30D20,30E20,30E25,47-01,47A10,47B01.
Frontcover4-Dgraphs,withcolorrepresentingthepolarangleoftheoutput,areof𝑓(𝑧)=𝑧2,
𝑓(𝑧)= (𝑧−1/2)(𝑧−𝑖/2) for𝑧∈𝔻,Γ(𝑧),𝑓(𝑧)=e𝑧,𝑓(𝑧)=cos𝑧,andtheRiemannzetafunction𝜁(𝑧).
(1−𝑧/2)(1+𝑖𝑧/2)
Foradditionalinformationandupdatesonthisbook,visit
www.ams.org/bookpages/text-71
LibraryofCongressCataloging-in-PublicationData
Names:Johnston,WilliamWilbur,1960–author.
Title:Thecalculusofcomplexfunctions/WilliamJohnston.
Description: Providence,RhodeIsland: AmericanMathematicalSociety,[2022]|Series: AMS/MAAtext-
books,2577-1205;volume71|Includesbibliographicalreferencesandindex.
Identifiers:LCCN2021044385|ISBN9781470465650(paperback)|ISBN9781470469016(ebook)
Subjects: LCSH:Functionsofcomplexvariables. |Calculus. |AMS:Functionsofacomplexvariable–In-
structionalexposition(textbooks,tutorialpapers,etc.) pertainingtofunctionsofacomplexvariable. |
Functionsofacomplexvariable–Seriesexpansionsoffunctionsofonecomplexvariable–Powerse-
ries(includinglacunaryseries)inonecomplexvariable.|Functionsofacomplexvariable–Geometric
functiontheory–Generaltheoryofconformalmappings. |Functionsofacomplexvariable–Entire
andmeromorphicfunctionsofonecomplexvariable,andrelatedtopics–Entirefunctionsofonecom-
plexvariable,generaltheory.|Functionsofacomplexvariable–Miscellaneoustopicsofanalysisinthe
complexplane–Integration,integralsofCauchytype,integralrepresentationsofanalyticfunctionsin
thecomplexplane. |Functionsofacomplexvariable–Miscellaneoustopicsofanalysisinthecomplex
plane–Boundaryvalueproblemsinthecomplexplane.|Operatortheory–Instructionalexposition(text-
books,tutorialpapers,etc.) pertainingtooperatortheory. |Operatortheory–Generalreferenceworks
(handbooks,dictionaries,bibliographies,etc.)pertainingtooperatortheory–Spectrum,resolvent.
Classification:LCCQA331.7.J642022|DDC515/.9–dc23/eng/20211105
LCrecordavailableathttps://lccn.loc.gov/2021044385
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InmemoriamPatriciaDunkelJohnston
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