Fourier Series Approximator

Math Calculus • Infinite Series and Sequences

Written by STEM Calculators Team Published December 30, 2025 Updated February 24, 2026

6. Fourier Series Approximator

Computes the partial Fourier series for a periodic function \(f(x)\) with period \(T\): \(S_N(x)=\dfrac{a_0}{2}+\sum_{n=1}^{N}\big(a_n\cos(n\omega_0 x)+b_n\sin(n\omega_0 x)\big)\), where \(\omega_0=\dfrac{2\pi}{T}\). Includes a zoomable, pannable plot and an optional “auto \(N\) for \(\varepsilon\)” helper.

Inputs

Function \(f(x)\)

Variable: x. Constants: pi, e. Supported: + − * / ^, parentheses, sin cos tan, asin acos atan, ln/log, sqrt, abs, exp, sgn.
Piecewise helper: pw(c, a, b) returns a if c is true, else b. Conditions: lt, le, gt, ge, eq, ne (return 1 or 0).

Period \(T\)

Accepts expressions like 2*pi, pi/2, 6.283.

Terms \(N\)

Computes coefficients up to \(n=N\) (numerical Simpson sampling).

Plot \(x_{\min}\)

Plot \(x_{\max}\)

Check at \(x_0\) (optional)

\(\varepsilon\) helper (optional)

Heuristic: find smallest \(N\le 300\) with \(|S_N(x_0)-S_{N-1}(x_0)|\le \varepsilon\).

Options

Plot \(f(x)\) Plot \(S_N(x)\) Shade \(|f-S_N|\) Coefficient table Show steps Auto-fit after solve Legend Animate \(N\) Prefer math.js (if available)

Integration subintervals

Even number required (Simpson).

Presets

Click a preset to load & compute.

Ready

Graph

Drag to pan • wheel/pinch to zoom. Plots periodic extension of \(f(x)\) and the partial sum \(S_N(x)\).

x: 0, y: 0, zoom(px/unit): 60

Results & steps

Enter inputs, then click Calculate.

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Theory: Fourier Series Approximator

Fourier series let you represent a periodic function as a sum of sines and cosines. The calculator computes the partial sum \(S_N(x)\) and compares it to the periodic extension of \(f(x)\).

1) Periodic functions and the fundamental frequency

A function \(f(x)\) is periodic with period \(T>0\) if \[ f(x+T)=f(x)\quad \text{for all }x. \] The fundamental angular frequency is \[ \omega_0=\frac{2\pi}{T}. \] In the calculator, if you enter a function that is not already periodic, it is interpreted as one period and then extended periodically.

Important: If the function has jumps or sharp corners, you’ll typically need a larger \(N\) to get a good approximation.

2) Fourier series (real form)

The real Fourier series of a \(T\)-periodic function is

\[ f(x)\sim \frac{a_0}{2}+\sum_{n=1}^{\infty}\left(a_n\cos(n\omega_0 x)+b_n\sin(n\omega_0 x)\right). \]

The coefficients are computed over any interval of length \(T\). A common choice is \([-T/2,\,T/2]\):

\[ a_0=\frac{2}{T}\int_{-T/2}^{T/2} f(x)\,dx, \qquad a_n=\frac{2}{T}\int_{-T/2}^{T/2} f(x)\cos(n\omega_0 x)\,dx, \qquad b_n=\frac{2}{T}\int_{-T/2}^{T/2} f(x)\sin(n\omega_0 x)\,dx. \]

The calculator displays the coefficient table \((a_n, b_n)\) and builds the partial sum \(S_N(x)\).

3) What the calculator plots

\(f(x)\) (periodic): the function is wrapped into one period (conceptually “tile the period forever”).
\(S_N(x)\): the Fourier partial sum using the first \(N\) harmonics.
Shaded error (optional): the region between \(f(x)\) and \(S_N(x)\) to visualize \(|f-S_N|\).

Even if your original formula is only defined nicely on one interval, the Fourier series approximates the periodic extension of that interval.

4) Convergence and what to expect

Fourier series convergence depends on smoothness:

If \(f\) is smooth, coefficients typically decay quickly and \(S_N(x)\) converges fast.
If \(f\) has corners or jumps, coefficients decay slowly and convergence is much slower.

At a jump discontinuity, the Fourier series converges to the midpoint of the left/right limits:

\[ \lim_{N\to\infty} S_N(x_0)=\frac{f(x_0^-)+f(x_0^+)}{2}. \]

5) Gibbs phenomenon (overshoot near jumps)

For functions with jump discontinuities (like a square wave), partial sums show oscillations near the jump. As \(N\) increases, the oscillation region becomes narrower, but the peak overshoot does not fully disappear (it approaches a fixed fraction of the jump size). This is the Gibbs phenomenon.

This is not a bug: it is expected Fourier-series behavior.

6) How coefficients are computed in this tool (numerical integration)

Exact integrals are not always practical, so the calculator approximates the coefficient integrals using Simpson’s rule over one period. If you increase “integration subintervals”, you get better accuracy, especially for sharp transitions (but it may run slower).

\[ \int_{-T/2}^{T/2} g(x)\,dx \approx \frac{h}{3}\left[g(x_0)+4g(x_1)+2g(x_2)+\cdots+4g(x_{n-1})+g(x_n)\right]. \]

7) “How many terms for ε” helper (what it means)

The tool provides a practical heuristic to pick \(N\) at a chosen \(x_0\). It looks for the smallest \(N\le 300\) such that the next term’s effect on the partial sum is small:

\[ |S_N(x_0)-S_{N-1}(x_0)|\le \varepsilon. \]

This is not a rigorous global error bound (Fourier series don’t have a simple universal remainder bound like Taylor), but it’s a useful indicator of whether the partial sums at \(x_0\) have “settled”.

Near discontinuities, this test can be misleading because Gibbs oscillations persist; choose \(x_0\) away from jumps.

8) Piecewise input syntax

Use pw(condition, a, b) to define piecewise expressions:

pw(c, a, b) returns \(a\) if \(c \neq 0\), else \(b\).
Conditions: lt(u,v), le(u,v), gt(u,v), ge(u,v), eq(u,v), ne(u,v) return 1 (true) or 0 (false).

Example (square wave-like):

\[ \texttt{pw(lt(abs(x),1),1,-1)}. \]

Frequently Asked Questions

What does the Fourier series approximator compute?

It computes a finite Fourier partial sum S_N(x) that approximates a T-periodic function using sine and cosine harmonics. The coefficients a_n and b_n are estimated numerically from integrals over one period.

How are the Fourier coefficients calculated in this tool?

The calculator approximates the required integrals with Simpson’s rule over an interval of length T. Increasing the integration subintervals can improve accuracy, especially for sharp corners or jump-like behavior.

What does the epsilon helper for N mean at x0?

It finds the smallest N (up to the tool limit) such that the change in the partial sum at x0 is small: |S_N(x0)-S_{N-1}(x0)|<=epsilon. This is a practical indicator at that point, not a guaranteed global error bound.

Why do I see overshoot near jumps even when N is large?

Fourier partial sums can show persistent oscillations near discontinuities, known as the Gibbs phenomenon. As N increases the oscillations narrow, but the peak overshoot does not fully disappear.

How do I enter a piecewise function for one period?

Use pw(condition,a,b) where the condition is built from lt, le, gt, ge, eq, or ne and returns 1 or 0. The calculator interprets your input over one period and then extends it periodically.

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Frequently Asked Questions

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