In addition to being mathematically elegant, complex variables provide a powerful tool for solving problems that are either very difficult or virtually impossible to solve in any other way. Part I of this text provides an introduction to the subject, including analytic functions, integration, series, and residue calculus and also includes transform methods, ODEs in the complex plane, numerical methods and more. Part II contains conformal mappings, asymptotic expansions, and the study of Riemann–Hilbert problems. The authors also provide an extensive array of applications, illustrative examples and homework exercises. This book is ideal for use in introductory undergraduate and graduate level courses in complex variables.
One of my favourite books along with Bender & Orszag's Mathematical Methods. Accessible language, lots of relevant examples, it's definetely a pleasant read. I had a first look into it when struggling to understand multivalued functions. Examples together with theory always worked best for me, so finding this book was one of the first steps towards my understanding branch cuts and practical ways to redefine them for non-trivial functions.
A great book on complex analysis and its many applications throughout mathematics. Standard topics are covered, such as complex and analytic (holomorphic) functions, multivaluedness and Riemann surfaces, complex integration, infinite series and products, singularities, analytic continuation, the residue theorem and more. The applications discussed include the Fourier and Laplace transform, fluid dynamics, ODEs in complex variables, asymptotic analysis and Riemann-Hilbert problems (both scalar and matrix). While the book does not contain as many proofs as some other books on the topic, it is definitely not an easy book and requires active reading. I took a complex analysis course before reading the book, and reading it still gave me a lot. I recommend the book (not only) to anyone who is interested in the ways complex analysis is applied in other parts of mathematics.
Complex variable, In mathematics, a variable that can take on the value of a complex number. In basic algebra, the variables x and y generally stand for values of real numbers. The algebra of complex numbers (complex analysis) uses the complex variable z to represent a number of the form a + bi...