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Fourier Series Coefficients Calculator

a₀, aₙ and bₙ over any period, computed exactly by integration by parts when f is a polynomial and by Gauss–Legendre quadrature otherwise, with the truncated series plotted against f so the Gibbs overshoot is visible.

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Fourier Series Coefficients Calculator is free to use as often as you like, directly from this page. You will find it under Calculus, with Fast Fourier Transform (FFT) Calculator and Maclaurin Series Calculator for the neighbouring cases.

How to use it

  1. Open the tool — no signup or install needed.
  2. Enter your input or adjust the available options.
  3. Get your result instantly, then copy or download it.

Frequently asked questions

What is Fourier Series Coefficients Calculator?

a₀, aₙ and bₙ over any period, computed exactly by integration by parts when f is a polynomial and by Gauss–Legendre quadrature otherwise, with the truncated series plotted against f so the Gibbs overshoot is visible.

When would I actually use this?

Checking a derivative or an integral you worked out by hand, finding where a function turns, and getting a numeric answer when no closed form exists.

What is the most common mistake?

Integrating across a discontinuity as if it were not there. A numeric method will happily return a finite value for an integral that diverges — check the domain before trusting the number.

How is Fourier Series Coefficients Calculator different from Fast Fourier Transform (FFT) Calculator?

They sit next to each other but answer different questions. Fast Fourier Transform (FFT) Calculator: radix-2 Cooley–Tukey FFT of real or complex samples (3+4i, −2i): magnitude, phase, frequency bins and a bar plot, with Hann, Hamming and Blackman windows corrected for coherent gain and one-sided bins correctly doubled except DC and Nyquist. Pick whichever matches what you're starting from — both are free.

Is there a tool for the next step?

Maclaurin Series Calculator is the closest one after this: The Taylor series at 0 for twelve standard functions — sin, cos, eˣ, ln(1+x), √(1+x), arcsin, arctan and more — with exact fractions such as 1/120, the general term and the error at a chosen x.

What else is worth having open alongside it?

Taylor Series Calculator and Infinite Series Sum Calculator — they come up in the same task often enough to be worth a second tab.

Where do the figures come from?

Symbolic results are exact; numeric ones come from adaptive quadrature or a standard step method, and the tool reports which. Where both are available, compare them — a large gap means the problem is ill-conditioned.

Further reading

All guides →
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