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12.2 Fourier Series Trigonometric Series is orthogonal on the interval [ -p, p]. In applications, we are interested to expand a function f(x) defined on [-p, p] as a linear combination Fourier series of the function f Fourier coefficients of f
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12.2 Fourier Series Example: Fourier series Expand in a Fourier series
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12.2 Fourier Series Example: Fourier series Expand in a Fourier series
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Convergence of a Fourier Series f(x) is piecewise continuous on the interval [-p,p]; if f(x) is continuous except at a finite number of points in the interval and have only finite discontinuities at these points. piecewise continuous Theorem 12.2.1 Conditions for Convergence piecewise continuous on [-p,p] is a point of continuity. is a point of discontinuity. denote the limit of f at x from the right and from the left
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12.2 Fourier Series Example: Expand in a Fourier series Remark:
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Sequence of Partial Sums Example: 15 terms 25 terms
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Sequence of Partial Sums Example: 15 terms 25 terms 125 terms 1000 terms MATHEMATICA Plot[0.5+Sum[ (1-(-1)^n)*Sin[n x]/(n Pi),{n,1,1000}],{x,-Pi,Pi}];
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Periodic Extension Example: Consider the funciion Periodic extension of the function f
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12.2 Fourier Series Example: Consider the function Periodic extension of the function f
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12.2 Fourier Series Example: Consider the function Periodic extension of the function a Fourier series not only represents the function on the interval ( -p, p) but also gives the periodic extension of f outside this interval. 2p is the fundamental period
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Periodic Extension Example: Consider the funciion (A) (B) (C) Which one represents FS(x) ?
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