Chapter 5 Elementary Functions
How should complex-valued functions such as \(e^z\text{,}\) \(\log z\text{,}\) \(\sin z\text{,}\) and the like, be defined? Clearly, any responsible definition should satisfy the following criteria.
- The functions so defined must give the same values as the corresponding functions for real variables when the number \(z\) is a real number.
- As much as possible, the properties of these new functions must correspond with their real counterparts. For example, we would want \(e^{z_1+z_2}=e^{z_1}e^{z_2}\) to be valid regardless of whether \(z\) were real or complex.
These requirements may seem like a tall order to fill. There is a procedure, however, that offers promising results. It is to put the expansion of the real functions \(e^x\text{,}\) \(\sin x\text{,}\) and so on, as power series in complex form. We use this strategy in this chapter.
