Our objective in this chapter is to provide a fairly detailed treatment of the differential calculus of complex variables up through the Cauchy-Riemann equations. In Chapter 7 we extend the analysis to include the complex integral and Laurent series. Because complex variables are used so extensively in engineering and physics applications, our discussion here and in Chapter 7 is generally more detailed than it is in other subjects. The analysis of elementary complex functions provides a natural means of mapping certain two-dimensional regions into other two-dimensional regions - a powerful concept that is further explored in Chapter 7 in connection with steady-state heat conduction and fluid flow.
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