Computing Fourier Series of f(x) = x on [-π,π]

Real Analysis · Axiom Academy

EXAMPLE Computing the Fourier Series of f(x)=x on Exploit the odd symmetry to kill every cosine term, then nail the sine coefficients with integration by parts. Find the Fourier series of f(x)=x on the interval . Use the symmetry of the function and integration by parts to determine all of its coefficients. Partial sums of the series — S₁ (1 term), S₃ (3 terms), S₅ (5 terms) — closing in on the dashed line y = x. More terms sharpen the fit on (−π, π); the overshoot near the ends is the Gibbs phenomenon. Nicely done — you computed the full Fourier series of f(x)=x on . Here is what carried the work: Symmetry does half the job: f(x)=x is odd, so a_0=0 and a_n=0 for all — only the sine terms survive. Integration by parts handles : with u=x , the product splits cleanly into a boundary term plus a vanishing integral. The boundary term carries the sign: turns the result into the alternating . A bonus: setting recovers the Leibniz series . The same recipe works for any function on : check the symmetry first, then compute the surviving coefficients — here, integration by parts on the sine terms.

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