BASIC COMPLEX ANALYSIS OF ONE VARIABLE - math.iitb.ac.in

1 BASIC COMPLEX ANALYSISOF ONE VARIABLEA nant R. ShastriDepartment of MathematicsIndian Institute of Technology, BombayAnd the detailed exposition can no less obfuscate than the overly terse. constructioniiPrefaceEvery mathematics student has to learn COMPLEX ANALYSIS . Infact every mathematicsteacher should teach a course in COMPLEX ANALYSIS at least once. However, every mathe-matics teacher need not write a book on COMPLEX ANALYSIS , here is yetanother book on this topic and I offer no justification for book is intended as a text/reference book for a first course in COMPLEX ANALYSIS (of duration one year or two semesters) for M. Sc. students inIndian universities andinstitutes of technologies. It is suitable for students whohave learnt to deal with basicset theoretic and arguments.

2 I assume that the student has been exposed tobasic differential and integral calculus of one real VARIABLE . It is also desirable that thestudent is exposed to some calculus of two variables, though, strictly speaking this isnot necessary. I hope that as the course proceeds, the student acquires more and book is the outcome of the lecture notes for the courses that I have taught atour department to M. Sc. students. In our department we teachroughly the materialin the first seven chapters in the first semester with three hours of lecture and one hourtutorial per week, as a compulsory course. The last three chapters can be covered in aleisurely fashion in a semester with three hours a week, as anelective course. However,it is also possible to adopt a slightly different order of presenting this material.

3 Forinstance, material in chapter 3, can be postponed until you come to chapter 9. On theother hand, most of the material in chapter 8 can be covered immediately after have tried to keep a dialogue style as far as possible. Throughout the book, I havetried to remember my own difficulties as a student. A student who wants to learn fromthis book should try to answer the questions that are being asked from time to timeand then proceed. Also, she should pause and reflect every time phrases such as it isobvious , or it follows are used, to see whether this really is so. For instance, when I amsaying that something is obvious and it is not at all obvious to her, she should perhapsiiiivPrefacere-read the material just before or some relevant topic thathas been covered , comments which are inside square brackets are meant for students who areabove average or those who have a better background.

4 The subsequent materials donot depend upon them. Enough exercises have been included totake care of studentsof various calibre. Some of them have been marked with a star,not to discourage thestudent from trying it but to tell her that even if she does notget it at the first attempt,it is alright. Of course some of them may need several attempts and the students maynot have so much time to devote. In any case, hints and solutions are given to almostall exercises, so that the student can compare her answers with cannot be anything new to say in such a widely used elementary topic. Ihave freely borrowed materials from several standard books, a bibliography of whichhas been included. There are many other books worth mentioning as good books but Ihave not borrowed anything from them.

5 For instance, the choice of topics is almost asubset of those in Ahlfors s book, which has influenced me to agreat extent. Three otherbooks that I really liked are [P-L], [S] and [R]. I wrote down the material in section ,and later on was very happy to read similar exposition in Remmert s book, the Englishaddition of which had come to our library just the past five years, ever since I latexed my lecture notes, I have receivedimpetus and help from several quarters in converting these notes into a book. It allbegan with the typical query by several of my students: Sir, are you going to write abook?..why not? Prof. M. G. Nadkarni, Prof. M. S. Narasimhan, Prof. C. S. Seshadri,Prof. K. Varadarajan etc. have put in a lot of encouraging words. My wife Parvati wasfirst to go through those primitive notes and gave me a vague idea of the kind of taskbefore me.

6 Prof. R. R. Simha and Prof. R. C. Cowsik have offeredmany valuablecomments apart from encouragement. Indeed, apart from providing troubleshootingsuggestions with latex, Cowsik went through the pre-final version and has correctedseveral mathematical , grammatical and typographical errors. (The author is solelyresponsible for whatever errors still persist, despite this.) I am indebted to all thesepeople and many the summer of 1995, I spent three months at Inter-national Centre for Theo-retical Physics, Trieste, out of which about a month I spent enlarging and polishing thenotes. Because of the excellent facilities and environmentthere and the free time onegets, I could do a lot in that single month. Above all, it was quite enjoyable. I wouldlike to mention that the final conversion of the lecture notesinto the book form wasPrefacevcarried out under the Curricular Development Programme of my to the II-editionIt gives me great pleasure to place this thoroughly revised edition of my book, thoughsomewhat edition contains more than 470 pages as compared to 300 pages in the firstedition.

7 Besides bringing further clarity in the presentation by reorganizing the material,I have added quite a bit of new material such as the homotopy version of Cauchy stheorem, Runge s theorem and a whole chapter on periodic functions culminating intoproof of Picard theorems. Throughout I have added more exercises also. In few placesI have cut down some material as things I have said in the preface to the first edition is valid for this edition aswell. This edition contains enough material for a first course (one year or equivalentlytwo semesters) in COMPLEX ANALYSIS at M. Sc. level at Indian universities and of the new features of this edition is that part of the bookcan be fruitfully usedfor a semester course for Engineering students, who have a good calculus at the dependence tree to decide your route.

8 I have myself followed :( ) ; ; , , , ; , , ; ; sincere thanks are due to many colleagues, friends and students who have offeredcomments which has helped in improving this edition. I should especially mention R. , S. S. Bhoosnurmath, Gowri Navada, Goutam Mukherjee who have meticulouslygone through one or the other version of this manuscript listing out typos, raising objec-tions and offering suggestions. I had several opportunitiesto teach this material at theATM Schools of National Board for Higher education organized by various persons atvarious locations such as Bhaskaracharya Pratisthan, DEparment of Mathematics DelhiUniversity etc. I have benefited from interaction with college teachers and researchstudents who participated in these , I am solely responsible for whatever inaccuraciesstill persist and will hap-pily receive reports of any such and promise pos the corrections on my website.

9 Theprocess of revision had started right from the day I receivedthe author s copies of theI edition. It is said that you never finish writing a book but you abandon it at somestage. The same seems to apply to revising a book as R. ShastriSpring, 2010viiiPrefaceContentsPrefaceiii1 BASIC Properties of COMPLEX Arithmetic of COMPLEX Numbers .. *Why the Name COMPLEX .. Geometry of COMPLEX Numbers .. Sequences and Series .. Topological Aspects of COMPLEX Numbers .. Path Connectivity .. *Connectivity .. *The Fundamental Theorem of Algebra .. Miscellaneous Exercises to Ch. 1 .. 612 COMPLEX Definition and BASIC Properties .. Polynomials and Rational Functions .. Analytic Functions: Power Series .. The Exponential and Trigonometric Functions.

10 Miscellaneous Exercises to Ch. 2 .. 973 Cauchy Riemann Equations .. *Review of Calculus of Two Real Variables .. *Cauchy Derivative (Vs) Frechet Derivative .. *Formal Differentiation and an Application .. Geometric Interpretation of Holomorphy .. Mapping Properties of Elementary Functions .. Fractional Linear Transformations .. The Riemann Sphere .. Miscellaneous Exercises to Ch. 3 .. 1414 Contour Definition and BASIC Properties .. Existence of Primitives .. Cauchy-Goursat Theorem .. * Cauchy s Theorem via Green s Theorem .. Cauchy s Integral Formulae .. Analyticity of COMPLEX Differentiable Functions .. A Global Implication: Liouville .. Mean Value and Maximum Modulus .. Harmonic Functions.