Table Of ContentMathematical Methods
for
Physics and Engineering
Volume 1
Complex Calculus
Complex Calculus
Jorge L. deLyra
Copyright c 2018–2019 Jorge L. deLyra
(cid:13)
Feedback to the author via
[email protected]
Translated from the Portuguese
by
M. A. S. deLyra
Translation revised by the author
ISBN: 978-1793012050
Version 1.1 (15Feb19)
QFTL
JLDL
Dedicated to Cida,
for all her support,
all her patience and
her constant company
through all these years.
Preface
Theteaching of mathematics to physicsstudents is alongstandingprob-
lem. The main reason for this is that there is an enormous amount of
mathematics that must be learned by physicists. Unlike what may hap-
peninthecaseoftheprofessionalworkofamathematician, thephysicist
cannot be limited to studying a few specific areas of mathematics and,
instead of that, he must become able and fluent in a large number of
different and varied areas, since almost all mathematics has important
and, in fact, many times essential, uses in physics. Mathematics is for
the physicist, first and foremost, the language in which science is formu-
lated, thought, discussed, developed, recorded and transmitted.
On the other hand, since mathematics is just a part of the learning
necessities of a physicist, and not his central area of dedication, usually
thereis very little time available for the absorption of all this universeof
mathematics. This often leads to an excessively superficial approach to
the mathematics in its teaching to physics students, which also entails
significant losses, since mathematics is used in physics in an essential
wayasalanguage. Thisconflictbetweenextensionanddepthexistseven
within the rather restrictive scope of what is traditionally described as
the “mathematical methods of physics”, which usually already includes
a rather limited selection of topics, centered on those traditionally con-
sidered of greater relevance. This text intends to present an approach
thattriestotrack whatcouldbecalled the“middleway”inthisconflict.
This text is the result of several years of experience of the author
teaching the Mathematical Physics courses at the Physics Institute of
the University of Sa˜o Paulo. These are disciplines that are offered to
all the undergraduate students of the Institute, starting usually at the
beginning of the third year of the undergraduate curriculum. Occasion-
ally, good students may be able to take it already in the second term of
the second year. The presentation of the content assumes, as prelimi-
nary material, the basic physics introductory courses, as well as those of
vii
viii
calculus and linear algebra, that are usually taken during the first two
years. The typical level of preliminary preparation of the students was
oneofthefactorstakenintoconsiderationfortheelaborationofthetext,
and the level of abstraction adopted is compatible with a characteristi-
cally undergraduate course. One of the important objectives of the text
is to serve the mathematical needs of the more advanced physics courses
of the undergraduate physics curriculum.
Thecontent included,as wellastheapproachadopted, aretheresult
of an effort at updating and reorganizing these courses, which are tradi-
tionally part of the central core of all undergraduate physics curricula,
aiming at a more modern presentation, and also at an adaptation of
the content to the possibilities and limitations of a pair of one-semester
courses. The text is organized in several volumes. Experience shows
that it is possible to go through the contents of the first three in one
semester: “Complex Calculus”, “Fourier Transforms” and “Differential
Equations”. The volume on differential equations is limited to the use
of Cartesian coordinates. The subsequent volumes complete a second
semester, dedicated to the study of differential equations in cylindrical
and spherical coordinates, as well as to the introduction of the cor-
responding special functions, in two volumes: “Bessel Functions” and
“Spherical Harmonics”. The text is organized in the form of relatively
short chapters, each appropriate for exposition in one lecture.
The text is first and foremost a text on physics, which focuses at-
tention on the mathematical tools needed for the treatment of more
complex problems in physics. By and large, a considerable influence
of the books by Ruel V. Churchill on complex analysis, Fourier series
and boundary value problems, which are referred to in the bibliography,
is reflected in the text. The mathematics is treated seriously and as
completely as possible within the time limitations, but the motivation
is predominantly the application of this mathematics to physics. There-
fore, whenever possible the mathematics is approached in the context of
physics applications, with the widest possible use of examples and moti-
vations with a geometrical character, as well as of analogies with topics
of physics. For example, the study of analytic functions is developed
with the aid of an analogy with electrostatics and its scalar and vector
fields, by means of the use of tools of real calculus in two dimensions.
Many theorems are proven along the text, with varied levels of rigor
and generality; however, not all results presented are proven. The cri-
teria for the choice of what to prove or not include the effort and time
ix
required, themathematical complexity inherentto the topic, thelevel of
difficulty of the mathematics involved, and the usefulness of the proofs
in promoting a better understanding of the structures by the students.
Theemphasis in thepresentation of the mathematics is the clear exhibi-
tion and understanding of the structures and procedures involved, with
an algorithmic approach aiming at their utilization in the applications,
as well as its use as a language in physics, and not the aspects of rigor
and generality. The theorems of real analysis and linear algebra which
are assumed known, and which are used without proof, are explicited in
the text. Some of them are also discussed in an appendix.
Each chapter includes alist of proposedproblems, which have varied
levels of difficulty, including exercises and practice problems, problems
that complete and extend the material presented in the text, and some
longer and more difficult problems, which are presented as challenges to
the students. These longer and more difficult problems are marked as
“challenge problems”. Some of these challenge problems involve proofs,
but these proofs are not considered a central part of the material, in
the sense of what should be required in tests and exams. The intent of
theseproblemsis to give even tothebeststudentssomething non-trivial
to face and against which to test their abilities. The results of some of
the problems are used in later parts of the text, and such problems
are marked as “reference problems”. A high priority should be given
to such problems in recitation classes. There are complete solutions
available, detailed and commented, to all the problems proposed, which
are presented in separate volumes, organized according to the sequence
of corresponding chapters.
Although thetext itself does notinclude an explicitly computational
approach, the presentation is such that it allows for a relatively direct
connection with the algorithmic and computational aspects. For ex-
ample, the topic of Fourier transforms and series is developed from a
discrete realization on a finite lattice, which is directly applicable to cal-
culations of a computational nature. Starting from this structure, one
then takes the continuum limit in order to derive the usual structures
of the Fourier series and transforms, with various types of boundary
conditions. Due to the fact that it treats the mathematics with a good
level of precision, as well as with an appropriate level of rigor, keeping
at the same time a practical and geometrical character, this text can be
used both by physics students and by students of other areas, such as
applied mathematics and engineering.
x
This Volume
This volume is dedicated to the complex calculus. This is a more prac-
tical and less abstract version of complex analysis and of the study of
analytic functions. This does not mean that there are no proofs in the
text, since all the fundamental theorems are proved with a good level
of rigor. The text starts from the very beginning, with the definition of
complex numbers,and proceeds up to the study of integrals on the com-
plexplaneandonRiemannsurfaces. Thefactsandtheoremsestablished
here will beused routinely in all the subsequentvolumes of this series of
books. The development is based on an analogy with vector fields and
with electrostatics, emphasizing interpretations and proofs that have
a geometrical character. The approach is algorithmic and emphasizes
the representation of functions by series, with detailed discussion of the
convergence issues.
Acknowledgements
The author would like to thank the contributions of the several teach-
ing assistants that acted in the courses on Mathematical Physics under
his supervision, as well as to the many students that took the courses,
throughout the years of development of this text, and that in a way
played the role of guinea pigs during that development. Without such
ableandwillingtestsubjects,itwouldnothavebeenpossibletodevelop
the text to the considerable extent which was achieved.
Many students and teaching assistants contributed by finding errors
in the text and in the equations, both in the text itself and in the
solutions to the problems, but the main contribution of the students is
always to readand usethetext withacritical mind-set, as well as toask
intelligent and challenging questions, of which there were many. This
certainly contributed to add both quality and scope to the text.
