PART D
Complex Analysis
- CHAPTER 13 Complex Numbers and Functions. Complex Differentiation
- CHAPTER 14 Complex Integration
- CHAPTER 15 Power Series, Taylor Series
- CHAPTER 16 Laurent Series. Residue Integration
- CHAPTER 17 Conformal Mapping
- CHAPTER 18 Complex Analysis and Potential Theory
Complex analysis has many applications in heat conduction, fluid flow, electrostatics, and in other areas. It extends the familiar “real calculus” to “complex calculus” by introducing complex numbers and functions. While many ideas carry over from calculus to complex analysis, there is a marked difference between the two. For example, analytic functions, which are the “good functions” (differentiable in some domain) of complex analysis, have derivatives of all orders. This is in contrast to calculus, where real-valued functions of real variables may have derivatives only up to a certain order. Thus, in certain ways, problems that are difficult to solve in real calculus may be much easier to solve in complex analysis. Complex analysis is important in applied mathematics for three main reasons:
1. Two-dimensional potential problems can be modeled and solved by methods of analytic functions. This reason is the real and imaginary parts of analytic functions satisfy Laplace's equation in two real variables.
2. Many difficult integrals (real or complex) that appear in applications can be solved quite elegantly by complex integration.
3. Most functions in engineering mathematics are analytic functions, and their study as functions of a complex variable leads to a deeper understanding of their properties and to interrelations in complex that have no analog in real calculus.