Build a sine series for an odd periodic ramp
An odd ramp is a useful model for understanding how sine harmonics represent a signal with a jump in its periodic extension. State the interval and period before integrating. Here x is an angular variable in radians. The periodic extension of f(x) = x on −π < x < π has period 2π. Its odd symmetry removes the cosine coefficients and the mean term.
Worked practice example
Find the Fourier series of f(x) = x on −π < x < π.
| Step | Working |
|---|---|
| Coefficient to calculate | bₙ = (2/π)∫ from 0 to π x sin(nx) dx. |
| Integrate by parts | ∫x sin(nx) dx = −x cos(nx)/n + sin(nx)/n². |
| Apply the endpoints | The integral is −π(−1)ⁿ/n; therefore bₙ = 2(−1)ⁿ⁺¹/n. |
| Write the series | f(x) = 2[sin x − sin(2x)/2 + sin(3x)/3 − …], for −π < x < π. |
Check your method
At the jump in the periodic extension, the Fourier series converges to the midpoint of the left and right limits: (π + (−π))/2 = 0. The endpoint value is therefore different from simply substituting π into the original nonperiodic formula x. Finite partial sums can overshoot near jumps.
Try it yourself
For f(x) = 2x on the same interval, find b₁, b₂ and b₃.
Show the answer and method
Linearity doubles every coefficient: b₁ = 4, b₂ = −2 and b₃ = 4/3. The mean and cosine coefficients remain zero.
This written practice example was prepared for Jemshah revision. It is additional practice, not an official examination question or a transcript of the video.