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Complex Analysis with Applications [Grafakos] - mklabgr - 09-03-2026
Complex Analysis with Applications
Authors: Nakhlé H. Asmar & Loukas Grafakos
Publication date: 2018
Publisher: Springer Cham
Series:Undergraduate Texts in Mathematics
Complex Analysis with Applications is an undergraduate textbook designed primarily for a one-semester course in complex analysis, particularly for upper-level mathematics students, while remaining accessible to students in engineering and the applied sciences. Its central philosophy is to develop rigorous complex analysis alongside concrete applications rather than treating applications as an afterthought. The authors begin with complex numbers, elementary complex functions and mappings, then develop analytic functions and the Cauchy–Riemann equations before proceeding to contour integration and the major results surrounding Cauchy's integral theorem and integral formula.
The middle of the book develops the core machinery of the subject: power series, sequences and series of analytic functions, isolated singularities, Laurent expansions, and residue theory. Residues are then used to evaluate complex integrals and related real integrals. The final chapters move toward the geometric and applied side of complex analysis, covering harmonic functions, Laplace's equation and conformal mappings, including transformations such as Schwarz–Christoffel mappings. This makes the book especially useful for readers interested in how complex-variable techniques connect to applied mathematics, physics and engineering. [/url]
A particularly strong feature is its pedagogical design. The book contains many fully worked examples, a large collection of exercises ranging from routine problems to project-style questions, hints for more difficult exercises, and extensive graphical illustrations—385 black-and-white figures and four colour illustrations. Springer also provides solutions to selected exercises. The combination of rigorous proofs, visual intuition and applications makes it somewhat more approachable than many classical complex-analysis texts while still providing a substantial mathematical treatment of the subject.
Main topics
- Complex numbers and complex-valued functions
- Analytic and holomorphic functions
- Cauchy–Riemann equations
- Complex integration
- Cauchy's theorem and Cauchy's integral formula
- Power series and analytic-function series
- Singularities and Laurent series
- Residue theorem
- Harmonic functions and Laplace's equation
- Conformal mappings
- Schwarz–Christoffel transformations
Key takeaways
- Level: Upper-undergraduate complex analysis.
- Style: Rigorous, but strongly example- and application-oriented.
- Best feature: Theory and applications are developed together rather than separated.
- Suitable for: Mathematics students as well as engineering and applied-science students.
- Self-study: Quite suitable because of the worked examples, illustrations, exercises and available partial solutions.
[url=https://link.springer.com/book/10.1007/978-3-319-94063-2?utm_source=chatgpt.com]Springer — Complex Analysis with Applications