Sawtooth Wave Series

Fourier Analysis · Axiom Academy

EXAMPLE Sawtooth Wave Fourier Series Complete derivation showing the alternating sine series using integration by parts We will derive the complete Fourier series using symmetry analysis and integration by parts. Excellent work! You've completed the sawtooth wave Fourier series derivation. Here's what we learned: Symmetry Analysis: Recognizing odd symmetry immediately tells us that a₀ = 0 and all aₙ = 0, simplifying our work significantly. Integration by Parts: This technique is essential for Fourier series with polynomial functions. Choose u = x and dv = sin(nx)dx for efficient computation. Boundary Behavior: The cosine terms at the boundary dominate, while the integral of cos(nx) vanishes due to periodicity. Alternating Series: The (-1)ⁿ⁺¹ pattern creates the characteristic sawtooth shape through constructive and destructive interference of harmonics. Convergence: The series converges pointwise to the function except at discontinuities (at ±π), where it converges to the average of left and right limits (zero).

This is the written version of the interactive lesson above. See the full Fourier Analysis course.