Areas of Applied Mathematics

All calculus textbooks start withf(x):fis a function of one (real) variablex. Topics covered include limits, continuity, differentiation, and integration, with the associated notation, such as df/dx=f'(x) and$\int _{a}^{b}f(x)dx$. It is also usual to include a discussion of infinite sequences and series. The rigorous treatment of all these topics constitutesreal analysis. Complex analysis starts with the following question. What happens if we replacexbyz=x+ iy, wherexandyare two independent real variables and i$=\sqrt{-1}?$Answering this question leads to rich new fields of mathematics; we