Variable Compleja: A Comprehensive Guide by Polya and Latta
Variable compleja, or complex variables, is a branch of mathematics that deals with functions of complex numbers. Complex numbers are numbers that have both a real and an imaginary part, such as 2 + 3i, where i is the square root of -1. Complex numbers can be represented on a plane, where the horizontal axis is the real part and the vertical axis is the imaginary part. This plane is called the complex plane.
Functions of complex variables are functions that take complex numbers as inputs and produce complex numbers as outputs. For example, f(z) = z^2 + 1 is a function of complex variables, where z is a complex number. Functions of complex variables have many applications in physics, engineering, and other fields, such as electromagnetism, fluid dynamics, quantum mechanics, and signal processing.
One of the best books to learn variable compleja is the book by George Polya and Gordon Latta, which was first published in 1974. This book is a classic textbook that covers the basic concepts and methods of variable compleja in a clear and rigorous way. The book also includes many examples and exercises to help the reader understand and apply the theory.
In this article, we will give an overview of the book by Polya and Latta and summarize its main topics and features. We will also provide some reviews and recommendations for anyone who wants to learn variable compleja from this book.
Overview of the Book by Polya and Latta
The book by Polya and Latta is divided into 12 chapters, each covering a different topic of variable compleja. The chapters are:
Complex numbers: This chapter introduces the concept of complex numbers and their properties, such as addition, subtraction, multiplication, division, conjugation, modulus, argument, polar form, exponential form, etc.
Complex functions: This chapter defines the concept of complex functions and their properties, such as domain, range, graph, limit, continuity, derivative, etc.
Elementary functions: This chapter discusses some of the most common complex functions, such as exponential function, logarithmic function, trigonometric function, hyperbolic function, inverse function, etc.
Complex integration: This chapter introduces the concept of complex integration and its properties, such as linearity, independence of path, Cauchy's theorem, Cauchy's integral formula, Morera's theorem, etc.
Series and product development: This chapter deals with the representation of complex functions by series and products, such as power series, Taylor series, Laurent series, infinite products, etc.
Singularities and residues: This chapter studies the behavior of complex functions near singular points and their classification into removable singularities, poles, essential singularities and branch points. It also introduces the concept of residue and its applications to evaluate integrals.
Conformal mapping: This chapter explores the concept of conformal mapping and its properties, such as angle preservation, local inversion, Schwarz lemma, Riemann mapping theorem, etc. It also discusses some examples of conformal mappings and their applications.
Harmonic functions: This chapter relates complex functions to harmonic functions and shows how to find harmonic conjugates and harmonic polynomials. It also introduces the Dirichlet problem and its solution by Poisson's integral formula.
The Schwarz-Christoffel transformation: This chapter presents a special type of conformal mapping that transforms the upper half-plane into a polygonal region. It shows how to find the parameters of the transformation and how to use it to solve boundary value problems.
The gamma function: This chapter introduces the gamma function and its properties, such as recurrence relation, reflection formula, duplication formula, etc. It also discusses some applications of the gamma function to evaluate integrals and sums.
The zeta function: This chapter introduces the zeta function and its properties, such as analytic continuation, functional equation, zeros and poles. It also discusses some applications of the zeta function to number theory and physics.
The prime number theorem: This chapter proves the prime number theorem using complex analysis techniques. The prime number theorem states that the number of primes less than or equal to x is asymptotic to x/log x as x tends to infinity.
The book also includes an appendix that contains some useful formulas and tables for variable compleja. The book is suitable for undergraduate and graduate students who want to learn variable compleja from a classical perspective. The book assumes some background knowledge in calculus and linear algebra. The book is written in a clear and rigorous style with many examples and exercises to help the reader understand and apply the theory.
Reviews and Recommendations of the Book by Polya and Latta
The book by Polya and Latta has received positive reviews and recommendations from many readers and experts who have used it to learn variable compleja. In this article, we will share some of these reviews and recommendations and highlight the strengths and weaknesses of the book.
One of the reviews of the book is by Alvin M. White, who wrote in SIAM Review that the book is \"a very readable introduction to complex variables\" that \"covers all the standard topics\" and \"contains many interesting examples and exercises\". He also praised the book for its \"clear exposition\" and \"rigorous proofs\". He noted, however, that the book is \"not intended as a reference work\" and that it \"does not cover some of the more advanced topics\" such as Riemann surfaces, elliptic functions, etc.
Another review of the book is by R. E. Barnhill, who wrote in International Journal for Numerical Methods in Engineering that the book is \"a well-written introduction to complex variables\" that \"provides a solid foundation for further study\". He also complimented the book for its \"many illustrative examples\" and \"numerous exercises\". He pointed out, however, that the book is \"somewhat dated\" and that it \"does not include some of the more recent developments\" such as numerical methods, boundary element methods, etc.
A recommendation of the book is by a user on Google Books, who wrote that the book is \"the best introductory text I know for complex analysis\" that \"provides a solid foundation and is written in a very clear and pedagogical manner\". He also recommended two other books for further study: A Course of Modern Analysis by E. T. Whittaker and G. N. Watson for a classical foundation in special function theory and Complex Variables by Mark J. Ablowitz and Athanassios S. Fokas for everything else.
These are some of the reviews and recommendations of the book by Polya and Latta. As you can see, most readers and experts agree that the book is a good introduction to variable compleja that covers the basic concepts and methods in a clear and rigorous way. The book also includes many examples and exercises to help the reader understand and apply the theory. The main drawbacks of the book are that it is not a comprehensive reference work and that it does not include some of the more advanced or recent topics in variable compleja. The book is suitable for undergraduate and graduate students who want to learn variable compleja from a classical perspective.
Another place where you can buy the book is eBay.com, which is an online platform that connects buyers and sellers of various products. You can find the book by searching for its title or ISBN number (0471693308) on the website. You can also see the product details, such as condition, price, shipping options, seller feedback, etc. You can also bid on the book or buy it now if it is available.
The price of the book on eBay.com depends on the condition and availability of the book and the seller's preferences. As of January 2022, the price of a used hardcover copy of the book ranges from $29.99 to $35.60, plus shipping costs. You can also compare prices and offers from different sellers and choose the best deal for you.
A third place where you can buy the book is Google Books, which is a service that allows you to search and preview books online. You can find the book by searching for its title or ISBN number (0471693308) on the website. You can also see some information about the book, such as publisher, publication date, language, format, etc. You can also read some pages of the book for free.
To buy the book from Google Books, you need to click on the \"Get this book in print\" button and choose one of the options that are available. You can either buy a new or used copy of the book from an online bookstore or library, or you can find a local bookstore that sells the book near you. The price and availability of the book may vary depending on the option that you choose.
Other Formats of the Book by Polya and Latta
If you prefer to read the book by Polya and Latta in a different format than paperback or hardcover, you may wonder what other options are available. In this article, we will give you some information on other formats of the book and where to find them.
One of the other formats of the book is PDF, which is a digital file that can be viewed on a computer or a mobile device. You can find a PDF version of the book on Google Books, which is a service that allows you to search and preview books online. You can read some pages of the book for free on Google Books, but you cannot download or print the whole book. You can also buy a print copy of the book from Google Books if you want.
Another format of the book is DJVU, which is a compressed file that can be viewed on a computer or a mobile device. You can find a DJVU version of the book on Weebly, which is a website that hosts various files and documents. You can download the DJVU file for free from Weebly, but you need a special software or app to open it. You can also convert the DJVU file to PDF or other formats if you want.
A third format of the book is SIAM Review, which is a journal that publishes articles and reviews on mathematics and its applications. You can find a review of the book by Alvin M. White in SIAM Review, which was published in 1976. You can read the review online on ACM Digital Library, which is a platform that provides access to various publications and resources in computing and related fields. You need a subscription or an institutional access to read the full review on ACM Digital Library.
In this article, we have discussed the book by Polya and Latta on variable compleja, which is a branch of mathematics that deals with functions of complex numbers. We have explained what the book is, what it can do, how to use it, how to download it, what other users think about it, and what other formats are available. We hope that this article has been helpful and informative for you.
If you want to buy the book by Polya and Latta, you can visit one of the websites that we have mentioned, such as Amazon.com, eBay.com, Google Books, Weebly, or ACM Digital Library. You can also compare prices and offers from different sources and choose the best option for you. You can also read some reviews and recommendations from other readers and experts who have used the book to learn variable compleja.
Thank you for reading this article and we hope that you enjoy reading the book by Polya and Latta on variable compleja. b99f773239