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Chapter 7: Special Functions

Although we have a good theoretical understanding of limits now, we're still missing many of the most important functions of analysis, such as the the trigonometric functions. In Chapter 4 we introduced exponential functions and in Chapter 5 we used those to define logarithms. Now we'll expand the definition of exponentials to include complex exponentials and use those as the basis for defining the trigonometric functions. Some work is then called for to show that the functions we define really do have the familliar properties of the trigonometric functions. All of this relies on infinite series; specifically power series. So we start the chapter with an more general discussion of infinite series, leading in to power series. Later in the chapter, we'll use Taylor series to find power series which express a variety of other common functions.

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