Fourier Series and Integrals

One of the most common solution techniques applicable to linear homogeneous partial differential equation problems involves the use of Fourier series. A discussion of the methods of solution of linear partial differential equations will be the topic of the next chapter. In this chapter, a brief outline of Fourier series is given. The primary concerns in this chapter are to determine when a function has a Fourier series expansion and then, does the series converge to the function for which the expansion was assumed Also, the topic of Fourier transforms will be briefly introduced, as it can also provide an alternative approach to solve certain types of linear partial differential equations. 

In order to establish the conditions for a function to have a Fourier series expansion, the following definitions are necessary. 

A function is said to be piecewise continuous in an interval a x b if there exist finitely many points a = X X2 x = b, such that/(x) is continuous in Xj x Xj+i and the one-sided limits/(x/ ) and/(xylj.i) exist for ally = 1,2,3,. .., —1 [1 ]. Note that a function is piecewise continuous on the closed interval a x b, however continuity on the open interval a X b does not imply piecewise continuity there. For example, 

When a function/(x) is piecewise continuous on an interval a x b, the integral of /(x) from x = a to x = b always exists. That integral is the sum of the integrals affix) over the open subintervals on which/is continuous, that is 

If two functions/i and are piecewise continuous on an internal a x b, then there is a finite subdivision of the interval such that both functions are continuous on each closed subinterval whenever the functions are given their limiting values from the interior at the endpoints. This means that a linear combinatitHi Ci/i + 2/2 or the product /1/2 has the continuity on each subinterval and is, itself, piecewise continuous on a x h. As a consequence 

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Hanna, R. Fourier Series and Integrals of Boundary Value Frohlems, Wiley, New York (1982). 

DyM] Dym, H. and H. McKean, Fourier Series and Integrals (Probability and Matbematical Statistics, Vol. 14) Academic Press, San Diego, 1972. 

Carslaw, H. S. The Theory of Fourier Series and Integrals, 3ded., Dover, New York (1930). 

David L. Powers, Boundary Value Problems, Harcourt/Academic Press, New York, 1999. This book includes a 40-page chapter on Fourier series and integrals. 

Chapter 4 and Chapter 5 introduce Sturm-Liouville problems and Fourier series and integrals, respectively. These topics contain essential background... 

Dym H and McKean HP (1972) Fourier Series and Integrals. New York Academic Press. 

Kreyszig (2005) Advanced Engineering Mathematics (9th edition) by E. Kreyszig. John Wiley Sons. Chapter 11 is Fourier Series and Integrals. I find this book to be complete and well organized, and a useful reference. Some complain that it spoon-feeds the student others say it is difficult to understand. 

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