Zeigerpaare: die komplexe Fourier-Reihe als Zeigerkette
About this app
cₙe^{int} und c₋ₙe^{−int} drehen gegenläufig, ihre Imaginärteile heben sich auf: Die Spitze der Kette bleibt auf der reellen Achse und zeichnet Dreieck oder Sägezahn. Zweiseitiges Spektrum mit Betrag und Phase, N Paare, Gibbs an den Sprüngen.
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 complex Fourier series f(t) = Σ cₙ e^{int} can be read as a chain of rotating pointers: each term is an arrow of length |cₙ| that turns n times per period. For a real function, c₋ₙ is the mirror image of cₙ. Each pair cₙe^{int} + c₋ₙe^{−int} turns in opposite directions and together gives the real value aₙ cos nt + bₙ sin nt. That is why the tip of the whole chain stays on the real axis and draws the function. At jumps the partial sum overshoots by about nine percent of the jump height, even with many terms: the Gibbs phenomenon.
What the app shows
In the complex plane the chain of pointers from n = −N to N turns, with positive and negative n in two colours. The tip stays on the real axis, and its path runs along in the time trace below as a curve, next to the faint exact function. A two-sided spectrum shows the magnitudes |cₙ| with small phase pointers, and a magnifier shows a single pair with its parallelogram and real sum. You can choose sawtooth or triangle and the number of pairs N.
Try it yourself
Start with a single pair and increase N step by step: the triangle is almost perfect with a few pairs, the sawtooth needs many. Look at the jump of the sawtooth: the overshoot gets narrower with more pairs, but not smaller. Drag in the time trace to scrub time by hand, and follow in the magnifier how the imaginary parts of a pair cancel at every moment.
Frequently asked questions
- Why is the complex Fourier series real although it uses complex pointers?
- For real functions c₋ₙ is the complex conjugate of cₙ. The two pointers of a pair turn in opposite directions as mirror images, so their imaginary parts cancel at every moment. What remains is the real value aₙ cos nt + bₙ sin nt.
- What do negative frequencies mean in a Fourier series?
- A pointer with negative n simply turns the other way. It is the mirrored twin of the pointer with positive n, not a separate physical oscillation.
- What is the Gibbs phenomenon?
- At a jump, every partial sum of the Fourier series overshoots by about nine percent of the jump height. With more terms the overshoot becomes narrower, but it does not disappear.
- Why does the triangle converge faster than the sawtooth?
- The triangle is continuous, and its coefficients decay like 1/n². The sawtooth jumps, and its coefficients decay only like 1/n. So it needs many more pairs.
Subject: Fourier analysis
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