Polynomials, Rational Functions and Trigonometric Polynomials
In this chapter we want to illustrate the relevance of complex numbers in some elementary situations. After a brief discussion of the algebra of polynomials in Section 5.1, we prove the fundamental theorem of algebra and discuss solutions by radicals of algebraic equations in Section 5.2. In Section 5.3 we present Hermite’s decomposition formulas for rational functions, which are useful for the integration of rational functions, see Chapter 4 of [GM1]. Finally, in Section 5.4, we discuss some basic facts about trigonometric polynomials and, more generally, sums of sinusoidal signals. In particular we shall see that the spectrum of a signal completely identifies the signal itself, we shall prove the energy identity and present a sampling formula.
KeywordsRational Function Positive Root Fundamental Theorem Trigonometric Polynomial Sinusoidal Signal
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