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13. Consider the Fourier series of f(x) 5 - x, 0 |
| Answer» BARON Jean Baptiste Joseph Fourier (1768−1830) introduced the idea that any periodic function can be represented by a series of SINES and cosines which are harmonically related.Baron Jean Baptiste Joseph Fourier (1768−1830)Fig.1 Baron Jean Baptiste Joseph Fourier (1768−1830)To consider this idea in more detail, we need to introduce some DEFINITIONS and common terms.Basic DefinitionsA function f(x) is said to have period P if f(x+P)=f(x) for all x. Let the function f(x) has period 2π. In this case, it is enough to consider behavior of the function on the interval [−π,π].Suppose that the function f(x) with period 2π is ABSOLUTELY integrable on [−π,π] so that the following so-called Dirichlet integral is finite:π∫−π |f(x)|dx<∞;Suppose also that the function f(x) is a single valued, piecewise continuous (must have a finite number of jump discontinuities), and piecewise monotonic (must have a finite number of maxima and minima).If the conditions 1 and 2 are satisfied, the Fourier series for the function f(x) exists and converges to the given function (see also the Convergence of Fourier Series page about convergence conditions.)At a discontinuity x0, the Fourier Series converges tolimε→0 12[f(x0−ε)−f(x0+ε)].The Fourier series of the function f(x) is given byf(x)=a02+∞∑n=1 {ancosnx+bnsinnx},where the Fourier coefficients a0, an, and bn are defined by the integralsa0=1ππ∫−π f(x)dx,an=1ππ∫−π f(x)cosnxdx,bn=1ππ∫−π f(x)sinnxdx.Sometimes alternative forms of the Fourier series are used. Replacing an and bn by the new variables dn and φn or dn and θn, wheredn=√a2n+b2n,tanφn=anbn,tanθn=bnan,we can write:f(x)=a02+∞∑n=1 dnsin(nx+φn)orf(x)=a02+∞∑n=1 dncos(nx+θn).Fourier Series of Even and Odd FunctionsThe Fourier series expansion of an even function f(x) with the period of 2π does not involve the terms with sines and has the form:f(x)=a02+∞∑n=1 ancosnx,where the Fourier coefficients are given by the formulasa0=2ππ∫0 f(x)dx,an=2ππ∫0 f(x)cosnxdx.Accordingly, the Fourier series expansion of an odd 2π-periodic function f(x) CONSISTS of sine terms only and has the form:f(x)=∞∑n=1 bnsinnx,where the coefficients bn arebn=2ππ∫0 f(x)sinnxdx.Below we consider expansions of 2π-periodic functions into their Fourier series, assuming that these expansions exist and are convergent. | |