Complex Variables for Engineers with Mathematica

177 Pages·2022·6.145 MB·English

by  Seiichi Nomura

Synthesis Lectures on
Mechanical Engineering
Seiichi Nomura
Complex Variables
for Engineers with
Mathematica
Synthesis Lectures on Mechanical
Engineering
Thisseriespublishesshortbooksinmechanicalengineering(ME),theengineeringbranch
that combines engineering, physics and mathematics principles with materials science to
design,analyze,manufacture,andmaintainmechanicalsystems.Itinvolvestheproduction
and usage of heat and mechanical power for the design, production and operation of
machines and tools. This series publishes within all areas of ME and follows the ASME
technical division categories.
Seiichi Nomura
Complex Variables
for Engineers
with Mathematica
SeiichiNomura
UniversityofTexasatArlington
Arlington,TX,USA
ISSN2573-3168 ISSN2573-3176 (electronic)
SynthesisLecturesonMechanicalEngineering
ISBN978-3-031-13066-3 ISBN978-3-031-13067-0 (eBook)
https://doi.org/10.1007/978-3-031-13067-0
©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG
2022
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Preface
Complexvariabletheoryisoneofthemostimportantsubjectsinmathematics,andmany
excellent textbooks have been written. Complex variable theory has not changed over
centuries. As this is a classical branch in mathematics, there seems hardly any need for
yet another book on complex variables.
Complex variable theory is attractive for engineers as it offers elegant approaches for
certaintypesofdifferentialequationsinengineeringincludingheattransfer,solidmechan-
icsandfluidmechanics.However,agapexistsbetweenbookswrittenbymathematicians
and books written by engineers in their specific fields. Naturally, mathematicians tend
to emphasize rigorousness and consistency while less emphasizing applications. On the
otherhand,bookswrittenbyengineersoftenjumpdirectlytothespecifictopicsassuming
thatthereadersalreadyhavesufficientbackgroundofcomplexvariablesandthepathway
from theory to application is not clearly elucidated. Therefore, those who want to learn
howcomplexvariabletheoryisutilizedintheirfieldsareoftenfrustratedwhennecessary
background materials are not provided.
One of the motivations for writing this book is to close the gap above such that a
smooth transition from basic theory to application is accomplished. Although it is not
possible to cover all the topics in engineering exhaustively, the readers can at least find
the logic of how and why complex variables are effective for some of the engineering
problems.
Another motivation for writing this book is to demonstrate that the readers can take
advantage of a computer algebra system, Mathematica, to facilitate tedious algebra and
visualizecomplexfunctionssothattheycanfocusonprinciplesinsteadofspendingend-
lesshoursonalgebrabyhand.UnlikenumericaltoolssuchasMATLABandFORTRAN,
Mathematica can expand, differentiate, and integrate complex-valued functions symboli-
cally. Mathematica can be used as a stand-alone symbolic calculator or a programming
toolusingtheWolframLanguage.IfMathematicaisnotavailablelocally,WolframCloud
Basic can be used online as a free service to execute Mathematica statements.
This book is suitable for upper-level undergraduate students or graduate students in
STEM who are interested in the expeditious application of complex variables in their
selected fields. The only prerequisite is the undergraduate level of calculus.
v
vi Preface
The book consists of six chapters and Appendix. Chapters 1–5 discuss the fundamen-
tals of complex variables in a manner less rigorous but just necessary for the readers
to prepare for engineering application, which is discussed in Chap. 6. Chapter 1 is an
introductiontocomplexvariablesandvariouscomplexfunctionssuchasexponential,log-
arithmic and trigonometric functions transitioning from their real counterparts. Chapter 2
examines differentiability of complex-valued functions known as the Cauchy-Riemann
equationsandtheconceptofanalyticfunctionsthatarethemajorplayersincomplexvari-
ables. Chapter 3 discusses integral calculus of complex functions known as the Cauchy
theorem and Cauchy’s integral formula. The fundamental theorem of algebra can be
provenusingtheLiouvilletheoremdirectlyfromCauchy’sintegralformula.Chapter4is
aseriesexpansionofcomplexfunctionsknownastheTaylorandLaurentseries.Thecon-
cept of analytic continuation is used to show a stunning identity 1+2+3+... = − 1
12
known as the Ramanujan Summation. Chapter 5 discusses the residue theorem and its
applicationstoimproperintegrals.Thistopicusedtobeoneofthehighlightsofcomplex
variables, but it is of less importance today as symbolic software including Mathematica
can automatically carry out many integrations. Chapter 6 is the most important chapter
in the book and shows how complex variables can be applied to real engineering prob-
lems. Topics are selected from heat transfer, solid mechanics and fluid mechanics and
conformal mapping as well as a series expansion method are explained. Appendix is an
introduction to Mathematica essentials. Each chapter has sample Mathematica codes to
help understand the subject. Just like the best way to learn a programming language is
to actually type the program yourself, I suggest that the readers enter the Mathematica
commands in the book themselves to feel how complex variables work in action.
IcannotdenythatthisbookwasinfluencedbyProf.MichaelD.GreenbergoftheUni-
versityofDelawareandhispopulartextbook[8].Hisuniquestyleinthelectureandbook
leadmetobelievethatengineeringis,afterall,aninterpretationofmathematicsinchosen
fields. I would like to acknowledge my gratitude to Prof. Greenberg. I would also like to
thankmycolleagues,Dr.ErianArmaniosandDr.KathyHays-Stang,fortheirwillingness
to help, and Paul Petralia of Springer Nature for his support and encouragement.
Arlington, TX, USA Seiichi Nomura
Contents
1 FunctionsofComplexVariables ....................................... 1
1.1 Complex Numbers ............................................... 1
1.2 Complex Plane .................................................. 4
1.3 Complex-Valued Functions ........................................ 8
1.3.1 Exponential Function, ez .................................... 9
1.3.2 Trigonometric Functions .................................... 10
1.3.3 Logarithmic Function, log z ................................. 13
1.3.4 Branch Cut and Branch Points ............................... 15
1.4 Numerics ....................................................... 19
1.5 Problems ........................................................ 21
2 CalculusofFunctionsofComplexVariables ............................ 23
2.1 Differentiability (Cauchy-Riemann Equations) ....................... 23
2.1.1 Cauchy-Riemann Equations ................................. 24
2.1.2 Alternative Form of Cauchy-Riemann Equations ............... 27
2.1.3 Harmonic Functions ........................................ 28
2.1.4 Uniqueness of Analytic Functions ............................ 30
2.2 Problems ........................................................ 32
3 IntegrationsofFunctionsofComplexVariables ......................... 35
3.1 Integral Calculus ................................................. 35
3.2 Cauchy’s Theorem ............................................... 38
3.2.1 Morera’s Theorem ......................................... 41
3.3 Cauchy’s Integral Formula ........................................ 42
3.3.1 Contour Integral of zn ..................................... 42
3.3.2 Cauchy’s Integral Formula .................................. 43
3.3.3 Generalized Cauchy’s Integral Formula ....................... 44
3.3.4 Liouville’s Theorem ........................................ 48
3.3.5 Fundamental Theorem of Algebra ............................ 51
3.4 Problems ........................................................ 52
vii
viii Contents
4 SeriesofComplexVariableFunctions .................................. 53
4.1 Taylor Series .................................................... 53
4.1.1 Taylor Series of f(z) About z =a ......................... 54
4.1.2 Analytic Continuation ...................................... 60
4.1.3 Can We Prove 1+2+3+4+···=− 1 ? .................. 63
12
4.2 Laurent Series ................................................... 64
4.3 MathematicaCode ............................................... 72
4.4 Problems ........................................................ 73
5 Residues ............................................................. 75
5.1 Types of Singularities ............................................. 75
5.2 Residues ........................................................ 78
5.2.1 Definition of Residues ...................................... 78
5.2.2 Calculation of Residues ..................................... 78
5.3 Residue Theorem ................................................ 82
5.3.1 Residue at Infinity ......................................... 83
5.4 Application of Residue Theorem to Certain Integrals ................. 86
5.4.1 First Type ................................................. 87
5.4.2 Second Type .............................................. 89
5.4.3 Third Type ................................................ 97
5.4.4 Mathematica Code ......................................... 101
5.5 Problems ........................................................ 103
6 ApplicationstoEngineeringProblems .................................. 105
6.1 Conformal Mapping .............................................. 105
6.1.1 Solving Laplace Equation by Conformal Mapping ............. 107
6.1.2 Bilinear (Möbius) Transformation ............................ 112
6.2 General Solution to Laplace Equation ((cid:2)φ(x,y)=0) ................ 112
6.3 General Solution to Bi-harmonic Equation ((cid:2)(cid:2)φ(x,y)=0) .......... 114
6.4 Heat Conduction ................................................. 116
6.5 Solid Mechanics ................................................. 126
6.6 Fluid Mechanics ................................................. 137
6.7 Problems ........................................................ 150
Appendix:Introductionto Mathematica ................................... 153
References .............................................................. 169
Index ................................................................... 171
1
Functions of ComplexVariables
1.1 ComplexNumbers
Aformalwaytodefineacomplexnumber,z,istoassociatez withapairofrealnumbers,
x and y,as
z :(x,y). (1.1)
Addition,subtractionandmultiplicationbetweentwocomplexnumbers,z andz ,expressed
1 2
as
z :(x ,y ), z :(x ,y ) (1.2)
1 1 1 2 2 2
canbedefinedas
z ±z :(x ±x ,y ±y ), (1.3)
1 2 1 2 1 2
z ·z :(x x −y y ,x y +x y ). (1.4)
1 2 1 2 1 2 1 2 2 1
Divisionofz byz isdonebyfindingzthatsatisfies
1 2
z =z ·z. (1.5)
1 2
Bydefiningz :(x,y),Eq.(1.5)canbewrittenas
(x ,y )=(x x −y y,x y+y x). (1.6)
1 1 2 2 2 2
Equation(1.6)canbesolvedforx and yas
x x +y y x y −x y
x = 1 2 1 2, y = 2 1 1 2, (1.7)
x2+y2 x2+y2
2 2 2 2
©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2022 1
S.Nomura,ComplexVariablesforEngineerswithMathematica,
SynthesisLecturesonMechanicalEngineering,
https://doi.org/10.1007/978-3-031-13067-0_1