Formel-Verwandler: von der reellen zur komplexen Fourier-Reihe
About this app
Die Fourier-Reihe verwandelt sich Schritt für Schritt: Euler-Formel einsetzen, nach e^{inωt} sortieren, c_n einführen, negative Indizes — bis f(t) = Σ cₙe^{inωt}. Der Graph bleibt dabei unverändert; c_n und c_{−n} erscheinen als gespiegelte Zeiger.
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
The idea
The real Fourier series a₀/2 + Σ(aₙ cos nωt + bₙ sin nωt) can be written entirely in complex form: f(t) = Σ cₙ e^{inωt} with n running from −∞ to ∞. Getting there is pure rewriting: replace cosine and sine by Euler’s formulas, use 1/(2i) = −i/2, sort by e^{inωt} and e^{−inωt}, and abbreviate with c₀ = a₀/2, cₙ = ½(aₙ − i bₙ) and c₋ₙ = ½(aₙ + i bₙ). With negative indices the three parts become one single sum. Nothing changes in content: it is the same function.
What the app shows
At the top, the formula transforms in seven steps: shared building blocks glide to their new place, old ones fade and new ones appear. Equal parts carry equal colours. Below, a graph shows the signal (triangle, sawtooth or your own trigonometric polynomial), and it stays unchanged in every step. In the complex plane next to it, cₙ and c₋ₙ appear as mirror-image pointers, together with the point aₙ + i bₙ, halfway to which c₋ₙ lies.
Try it yourself
Step through the transformation one step at a time and watch where each building block goes. Choose the polynomial and move aₙ and bₙ: cₙ and c₋ₙ always move as mirror images in the real axis. Switch between triangle and sawtooth: for the even triangle the cₙ lie on the real axis, for the odd sawtooth on the imaginary axis.
Frequently asked questions
- How do you get from the real to the complex Fourier series?
- Replace cos and sin by Euler’s formulas, sort by e^{inωt} and e^{−inωt}, and abbreviate: c₀ = a₀/2, cₙ = ½(aₙ − i bₙ), c₋ₙ = ½(aₙ + i bₙ). With negative indices this gives f(t) = Σ_{n=−∞}^{∞} cₙ e^{inωt}.
- Is the complex Fourier series a different function?
- No. It is only a different way of writing the same real function. In the app the graph stays exactly the same in every step.
- How are cₙ and c₋ₙ related?
- For a real function they are complex conjugates: c₋ₙ is the mirror image of cₙ in the real axis. That is why the imaginary parts cancel in the sum.
- Why are the real coefficients called a and b?
- The expression aₙ − i bₙ looks like a complex number in Cartesian form a + ib. Indeed cₙ = ½(aₙ − i bₙ) is a complex number whose real and imaginary parts come from the real coefficients.
Subject: Fourier analysis
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