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

Differential Equations · Axiom Academy

EXAMPLE Finding the Fourier Series of f(x) = x on Compute the Fourier coefficients from the definitions and build the full sine-series representation Find the Fourier series of the function below on the interval , using the standard coefficient formulas for a_0 , a_n , and b_n . Excellent work — you've computed the full Fourier series for f(x) = x on . Here's what carries forward: Odd-function shortcut: since f(x) = x is odd, both a_0 and every a_n are zero without any integration — only sine terms can appear in the series. Integration by parts: computing b_n still takes work — needs u=x , , giving . The alternating pattern: the factor (-1)^ n+1 makes the coefficients alternate in sign — — characteristic of a discontinuous periodic extension. Convergence behavior: the partial sums converge to f(x)=x on the open interval , but at the endpoints the periodic extension jumps, so the series converges to 0 (the average of the left and right limits) there. Gibbs phenomenon: near the jump at , partial sums overshoot the function's value — an overshoot that persists (it doesn't shrink to zero) even as more terms are added. This same odd/even shortcut and integration-by-parts technique apply to any piecewise-smooth periodic function — Fourier series like this one are the foundation for solving PDEs with periodic forcing and for signal processing.

This is the written version of the interactive lesson above. See the full Differential Equations course.