Numerical Analysis Using MATLAB and Spreadsheets
This concise text provides complete, clear, and detailed explanations of the principal numerical analysis methods and well known functions used in science and engineering.
TABLE OF CONTENTS Numerical Analysis Using MATLAB and Spreadsheets
Chapter 6: Fourier, Taylor, and Maclaurin Series
This chapter is an introduction to Fourier and power series. We begin with the definition of sinusoids that are harmonically related and the procedure for determining the coefficients of the trigonometric form of the series. Then, we discuss the different types of symmetry and how they can be used to predict the terms that may be present. Several examples are presented to illustrate the approach. The alternate trigonometric and the exponential forms are also presented. We conclude with a discussion on power series expansion with the Taylor and Maclaurin series.
6.1 Wave Analysis
The French mathematician Fourier found that any periodic waveform, that is, a waveform that repeats itself after some time, can be expressed as a series of harmonically related sinusoids, i.e., sinusoids whose frequencies are multiples of a fundamental frequency (or first harmonic). For example, a series of sinusoids with frequencies 1 MHz, 2 MHz, 3 MHz, and so on, contains the fundamental frequency of 1 MHz, a second harmonic of 2 MHz, a third harmonic of 3 MHz, and so on. In general, any periodic waveform f( t) can be expressed as
or
where the first term a 0/ 2 is a constant, and represents the DC (average) component of f( t). Thus, if f( t) represents some voltage v( t), or current i( t), the term a 0/ 2 is the average value...
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