This is a set of lecture notes on introductory school algebra written for middle school teachers. Part of the undergraduate texts in mathematics book series utm. Elementary algebra, branch of mathematics that deals with the general properties of numbers and the relations between them. Complex variables covers topics ranging from complex numbers to point sets in the complex plane, elementary functions, straight lines and circles, simple and conformal transformations, and zeros and singularities.
A complex number can be visually represented as a pair of numbers a, b forming a vector on a diagram called an argand diagram, representing the complex plane. Obviously, there is a good deal of overlap, but some books deal primarily with one aspect or the other. The instantaneous velocity at the time moment t 0 is the derivative of x with respect. Complex numbers and their elementary properties functions of. The book begins with fundamentals, with a definition of complex numbers, their geometric representation, their algebra, powers and roots of complex numbers, set theory as applied to complex analysis, and complex functions and sequences. The answer to the question needs to be any point on the complex. The book is a real treasure trove of nontrivial elementary key concepts and applications of complex numbers developed in a systematic manner with a focus on problem solving techniques. Polar form and geometric interpretation mathematics. Symbolic expressions, transcription of verbal information into symbolic language, linear equations in one variable, linear equations in two variables and their graphs, simultaneous linear equations, functions and their graphs, linear. Much of the book goes to geometric applications, of course, but there are also sections on polynomial equations, trigonometry, combinatorics. The purely arithmetic theory of complex numbers as pairs of real numbers was introduced by w.
We also discussed fractional powers and their manyvaluedness. Re is the real axis, im is the imaginary axis, and i satisfies i2. Teachers using these notes would enhance the quality of their teaching and by implication, students success. Logarithmic functions are inverses of exponential functions, since logarithms are inverses of exponents.
He found a generalization of complex numbers, namely the quaternions cf. The plot above represents a path of this object as a function of time. Mathematical institute, oxford, ox1 2lb, july 2004 abstract this article discusses some introductory ideas associated with complex numbers, their algebra and geometry. Elements of the theory of functions dover books on.
Arithmeticoperations complex numbers can be added, subtracted, multiplied, and divided. More generally, it was proved at the end of the 19th century that any extension of the notion of number beyond. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. For which values of their arguments the product a b and the quotient a b are real numbers. Theory and technique is a book in a special category of influential classics because it is based on the authors extensive experience in modeling complicated situations and providing analytic solutions.
Functions of a complex variable are used to solve applications in various branches of mathematics, science, and engineering. Separate chapters discuss the fundamentals of complex analysis. Pdf bicomplex numbers and their elementary functions. There is associated with each complex number z x, y an additive inverse. In other words, it is the original complex number with the sign on the imaginary part changed. It is presented solely for those who might be interested. This book is destined to accompany many students making their way into a classical area of mathematics which represents the most fruitful example to date of the. Binney oxford university michaelmas term 2002 books.
The first section is a more mathematical definition of complex numbers and is not really required for understanding the remainder of the document. In introducing complex numbers, and the notation for them, this article brings together into one bigger picture some closely related elementary ideas like vectors and the exponential and trigonometric functions and their derivatives. Volume 2 and 3 theory of functions, parts i and ii table of contents. Of course, additionally, the proof serves to illustrate basic complex number techniques. An introduction to the theory of complex variables. We describe how to define elementary functions in such an algebra polynomials. Along with writing, it is a cornerstone of modern scientific and technological civilization. The importance of the statement and the corollary is underscored by the inclusion of a less elementary proof that employs complex numbers in a classic book on advanced geometry of plane curves. Elementary functions of a complex variable springerlink. We conclude these notes by defining three of the basic elementary functions that take complex arguments.
Complex analysiselementary functionsexponential functions. This includes a look at their importance in solving polynomial equations, how complex numbers add and multiply, and how they can be represented. A complex number is the fancy name for numbers with both real and imaginary parts. Elementary functions and complex numbers digital book, pp. The majority of problems are provided the majority of problems are provided with answers, detailed procedures and hints sometimes incomplete. It is useful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics. Cauchys theorem, taylors theorem, laurents theorem, contour integration, and miscellaneous theorems are also discussed.
We begin with the three representations of complex numbers. Complex numbers of the form x 0 0 x are scalar matrices and are called real complex numbers and are denoted by. We describe how to define elementary functions in such an algebra polynomials, exponential functions, and trigonometric functions as well as their inverse functions roots, logarithms, inverse trigonometric functions. In the preceding chapters we studied complex numbers from an algebraical point of view, coupled with geometrical interpretations, and this enabled us to arrive at a sensible and consistent definition of powers z r, where r is an integer. The totality of complex numbers or the set of complex numbers is usually denoted by the symbol c. Despite the historical nomenclature imaginary, complex numbers are. Complex analysis for mathematics and engineering, 3rd. A good book is one which aims to teach you the concept, and give you some challenging questions which in turn, will boost your understanding and confidence. The emphasis is on disentangling and explaining all the necessary concepts. The book is mainly devoted to complex numbers and to their wide applications. Complex numbers, like other kinds of numbers, are essen tially objects with which to perform calculations according to certain rules, and when this principle is borne in mind, the nature of complex numbers is no more mysterious than. Yet the book is far from being a mere history of function theory.
Complex analysiselementary functionslogarithmic functions. One of the most useful consequences stems from the elementary observation that the composition of two complex functions is also a complex function. Written in a classical style, it is in the spirit of the books by ahlfors and by saks and zygmund. The the picture refers to are only those on the complex unit circle.
Very long times ago people had no problem to accept the fact that an equation may have no solution. Bicomplex numbers and their elementary functions 65 subsequently, other authors have developed further the study of these objects, 2, 6, 8, 9. This digital book contains 9 lectures on practically everything stem students need to know about elementary functions and complex numbers to succeed in their courses. The complex plane and elementary functions springerlink. Theory of functions, parts i and ii dover books on. Core versions of cauchys theorem, and consequences 75 88. Here, a is called the real part 29 and \b\ is called the imaginary part 30. The instantaneous velocity at the time moment t 0 is the derivative of x with respect to time at the time. Lecture notes in elementary complex functions with computer. Here are some examples of complex numbers and their. Good books on complex numbers mathematics stack exchange.
Complex numbers, complex functions, elementary functions, integration, cauchys theorem. The rest of the partials are done in similar manner i. The instantaneous velocity at the time moment t 0 is the derivative of x with respect to time at the. In this paper we introduce the algebra of bicomplex numbers as a generalization of the field of complex numbers. A distinction needs to be made between purely geometric uses of complex numbers and uses in the theory of equations polynomials, rational functions, etc. Being designed for a onesemester course, it is much shorter than many of the standard texts. The material of this course is covered well in many texts on mathematical methods for science students, for example boas, mathematical methods in the physical sciences, 2nd ed. The lecture notes contain useful summaries cheat sheets and description of necessary study skills, including tips for preparing for tests and exam.
Complex numbers, like other kinds of numbers, are essen tially objects with which to perform. Algebra is fundamental not only to all further mathematics and statistics but to the natural sciences, computer science, economics, and business. Morgan commented in his book on the study and difficulties of mathematics, we. Functions of a complex variable and some of their applications. This is a textbook for an introductory course in complex analysis. Because no real number satisfies this equation, i is called an imaginary number. Solution using the elementary properties of complex numbers, it follows. This book is a revision of the sixth edition, published in 1996. It is important to note that any real number is also a complex.
The purpose of this book is to present a straightforward introduction to complex numbers and their properties. Complex variables and applications, seventh edition mariosuazo. Elements of the general theory of analytic functions section i. Even experts will find here few new or long forgotten gems, like eisensteins novel approach to the circular functions. Elementary functions complex numbers motivation for the complex. A key role, in this evolution, has been played by john ryan, who was probably the. In this manner, complex functions provide a rich lode of additional solutions to the twodimensional laplace equation, which can be exploited in a wide range of physical and mathematical applications. So, thinking of numbers in this light we can see that the real numbers are simply a subset of the complex numbers.
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