Exponential: f(x) = eˣ on (−π, π)

Set the period and define the function piece by piece, the way textbooks do. The tool finds a₀, aₙ, bₙ (exact formulas for polynomial pieces), draws partial sums, harmonics and phasors, and shows pointwise convergence.

Not a polynomial — the coefficients are computed numerically. At the period ends the series converges to cosh(π).

220 Hz

Function

Exact formulas

Original wave and series sum

Harmonics in the sum

Click a number to drop that harmonic from the sum and hear the difference.

Series terms

Spectrum

Convergence

Coefficient decay—
Max error at N—
Energy in N harmonics—
Gibbs overshoot—
Step-by-step derivation

Textbook problems

Every problem has its own page, with its own walkthrough, coefficients and charts.

Why this works

  • Sines and cosines at different frequencies are orthogonal over a full period: their mutual average is zero.
  • A Fourier coefficient is the projection of the signal onto one frequency. If that frequency is present, the projection is large; if it is not, it is nearly zero.
  • When those projections are added back together, each harmonic restores its part of the shape. More harmonics make the partial sum closer to the original wave.
  • Sharp corners and jumps require high frequencies. That is why square and sawtooth waves converge more slowly than a triangle wave.

How to read the visualization

  • The gray dashed line is the original periodic wave, and the blue line is the sum of the selected harmonics.
  • The lower rows show individual terms: each row is one sine wave with the frequency and amplitude used in the sum.
  • The phasors on the right show the same harmonics as rotating vectors. Their vertical projections add up to the current series value.
  • Increase the harmonic count and watch higher frequencies sharpen corners, peaks, and abrupt transitions.

Quick reference

A Fourier series represents a periodic function as a sum of sines and cosines: a constant part plus harmonics with frequencies 1, 2, 3, and so on. In engineering, it is the basis for analyzing audio, radio signals, vibration, images, and any repeating process.

This Fourier series visualizer shows how a complex periodic shape is assembled from simple sine waves. The original wave, partial sum, individual harmonics, and phasors are visible on one screen.

The tool is useful for understanding spectral analysis: square, sawtooth, and triangle waves have different harmonic sets and different convergence rates. The harmonic-count slider makes that visible without formal derivations.

All calculations run in the browser. Use the page as an interactive reference for Fourier series, harmonics, orthogonality, and the Gibbs effect.