© 2008, 2012 Zachary S Tseng E-2 - 1 Second Order Linear Partial Differential Equations Part II Fourier series; Euler-Fourier formulas; Fourier Convergence Theorem; Even and odd functions; Cosine and Sine Series Extensions; Particular solution of the heat conduction equation Fourier Series Suppose f is a periodic function with a period T = 2L. Then the Fourier series representation of f is a trigonometric series (that is, it is an infinite series consists of sine and cosine terms) of the form ∑∞=++=10sincos2)(nnnLxnbLxnaaxfππ Where the coefficients are given by the Euler-Fourier formulas: ∫−=LLmdxLxmxfLaπcos)(1, m = 0, 1, 2, 3, … ∫−=LLndxLxnxfLbπsin)(1, n = 1, 2, 3, … The coefficients a’s are called the Fourier cosine coefficients (including a0, the constant term, which is in reality the 0-th cosine term), and b’s are called the Fourier sine coefficients.© 2008, 2012 Zachary S Tseng E-2 - 2 Note 1: Thus, every periodic function can be decomposed into a sum of one or more cosine and/or sine terms of selected frequencies determined solely by that of the original function. Conversely, by superimposing cosines and/ or sines of a certain selected set of frequencies we can reconstruct any periodic function. Note 2: If f is piecewise continuous, then the definite integrals in the Euler-Fourier formulas always exist (i.e. even in the cases where they are improper integrals, the integrals will converge). On the other hand, f needs not to be piecewise continuous to have a Fourier series. It just needs to be periodic. However, if f is not piecewise continuous, then there is no guarantee that we could find its Fourier coefficients, because some of the integrals used to compute them could be improper integrals which are divergent. Note 3: Even though that the “=” sign is usually used to equate a periodic function and its Fourier series, we need to be a little careful. The function f and its Fourier series “representation” are only equal to each other if, and whenever, f is continuous. Hence, if f is continuous for −∞ < x < ∞, then f is exactly equal to its Fourier series; but if f is piecewise continuous, then it disagrees with its Fourier series at every discontinuity. (See the Fourier Convergence Theorem below for what happens to the Fourier series at a discontinuity of f .) Note 4: Recall that a function f is said to be periodic if there exists a positive number T, such that f (x + T ) = f (x), for all x in its domain. In such a case the number T is called a period of f. A period is not unique, since if f (x + T ) = f (x), then f (x + 2T ) = f (x) and f (x + 3T ) = f (x) and so on. That is, every integer-multiple of a period is again another period. The smallest such T is called the fundamental period of the given function f. A special case is the constant functions. Every constant function is clearly a periodic function, with an arbitrary period. It, however, has no fundamental period, because its period can be an arbitrarily small real number. The Fourier series representation defined above is unique for each function with a fixed period T = 2L. However, since a periodic function has infinitely many (non-fundamental) periods, it can have many different Fourier series by using different values of L in the definition above. The difference, however, is really in a technical sense. After simplification they would look the same.© 2008, 2012 Zachary S Tseng E-2 - 3 Therefore, technically at least, a Fourier series of a periodic function depends both on the function as well as its chosen period. Note 5: The definite integrals in the Euler-Fourier formulas can be found be integrating over any interval of length 2L. However, from −L to L is the convention, and is often the most convenient interval to use. Note 6: Since the Fourier coefficients are calculated by definite integrals, which are insensitive to the value of the function at finitely many points. Consequently, piecewise continuous functions of the same period that differ from each other at finitely many points (notably, at isolated discontinuities) per period will have the same Fourier series. Note 7: The constant term in the Fourier series, which has expression ∫∫−−=⋅=LLLLdxxfLdxxfLa)(21)0cos()(12120, is just the average or mean value of f (x) on the interval [−L, L]. Since f is periodic, this average value is the same for every period of f. Therefore, the constant term in a Fourier series represents the average value of the function f over its entire domain.© 2008, 2012 Zachary S Tseng E-2 - 4 Example: Find a Fourier series for f (x) = x, −2 < x < 2, f (x + 4) = f (x). First note that T = 2L = 4, hence L = 2. The constant term is one half of: 0)22(2122121cos)(1222220=−====−−−∫∫xdxxdxLxmxfLaLLπ The rest of the cosine coefficients, for n = 1, 2, 3, …, are ∫∫−−==222cos21cos)(1dxxnxdxLxnxfLaLLnππ 0)cos(40)cos(40212cos42sin2212sin22sin221222222222222=−+−+=+=−=−−−∫ππππππππππππnnnnxnnxnnxdxxnnxnnx Hence, there is no nonzero cosine coefficient for this function. That is, its Fourier series contains no cosine terms at all. (We shall see the significance of this fact a little later.)© 2008, 2012 Zachary S Tseng E-2 - 5 The sine coefficients, for n = 1, 2, 3, …, are ∫∫−−==222sin21sin)(1dxxnxdxLxnxfLbLLnππ ( ))cos(4)cos()cos(20)cos(40)cos(4212sin42cos2212cos22cos22122222222πππππππππππππππππnnnnnnnnnxnnxnnxdxxnnxnnx−=+−=−−−−−=+−=−−−=−−−∫ πππnevennnoddnnn4)1(,4,41+−==−==. Therefore,2sin)1(4)(11xnnxfnnππ∑∞=+−=.© 2008, 2012 Zachary S Tseng E-2 - 6 Figure: the graph of the partial sum of the first 30 terms of the Fourier series 2sin)1(4)(11xnnxfnnππ∑∞=+−=. Compare it against the graph of the actual function the series represents the function f (x) = x, −2 < x < 2, f (x + 4) = f (x), seen earlier.© 2008, 2012 Zachary S Tseng E-2 - 7 Example: Find a Fourier series for f (x) = x, 0 < x < 4, f (x + 4) = f (x). How will it be different from the series above?
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