Convergence of Periodic Fourier Series C.G. (2-01) original version C.G. (5-07) .mw version This Maple worksheet produces 2-D plots of a prescribed function defined on (NiQsJCUjcGlHISIiRiQ=) and its truncated periodic Fourier series. Re-initialize the workspace. restart; Define a function on (NiQsJCUjcGlHISIiRiQ=). Uncomment one of the functions below, or create one of your own. Use the Heaviside function for piecewise-defined functions. smooth periodic f := x -> exp(sin(x));

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smooth periodic, even # f := x -> exp(cos(x)); discontinuous angle function # f := x -> x * Heaviside(x); discontinuous sawtooth function, odd # f := x -> x; C0 with cusps # f := x -> sqrt(Pi^2-x^2); C0 with kinks # f := x -> Pi^2 - x^2; C1 # f := x -> ( Pi^2 - x^2 )^2; Set up to plot given function here in the worksheet. plotsetup(inline); Plot the given function over (NiQsJCUjcGlHISIiRiQ=). plot( f, -Pi..Pi, color=red );

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Periodic extension of given function. F := proc( x ) local s; s := x - 2 * Pi * floor((x+Pi)/(2*Pi)); f(s) end: Truncated Fourier series. S := x -> a[0] + sum( a[n]*cos(n*x)+b[n]*sin(n*x), n=1..N );

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Set up to plot in a separate window. plotsetup(window); Evaluate periodic Fourier coefficients, numerically, with 5-digit accuracy. Constant (DC) part. a[0] := evalf( int(f(x),x=-Pi..Pi)/(2*Pi), 5 );

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Truncate at this many modes: N := 1;

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Cosine and sine coefficients. for m from 1 to N do a[m] := evalf( int(f(x)*cos(m*x),x=-Pi..Pi) / Pi, 5 ): b[m] := evalf( int(f(x)*sin(m*x),x=-Pi..Pi) / Pi, 5 ): od;

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Plot given function and truncated series over slightly larger interval (extra 1/2-period at each end). Need to manually edit title (when change 'N'), legend (when change f(x)), and number of points (for higher frequencies). plot( [ F, S ], -2*Pi..2*Pi, discont=true, color=[red,blue], title="N = 1", legend=["f(x) = exp(sin(x))", "S(x)"], numpoints=128 );