Fourier Series
Any reasonable periodic function is a sum of sines and cosines. The coefficients are recovered by integrating against each one, which is what lets a system driven by a complicated periodic input be solved one frequency at a time.
What a learner can do afterwards
- Compute the first Fourier coefficients of a square or sawtooth wave
- Explain how orthogonality of the sines and cosines isolates each coefficient
- Say what the partial sums do near a jump discontinuity
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Any reasonable periodic function is a sum of sines and cosines. You stack them up: a0 over 2 plus a1 cosine x plus b1 sine x plus a2 cosine 2 x plus b2 sine 2 x, on and on. To get a0, average the function over one full period. To get each an, multiply by cosine n x and integrate over a period, then double and divide by the period length. Do the same trick with sine n x for each bn.
Take the square wave that is minus 1 on minus pi to 0 and plus 1 on 0 to pi, repeating every 2 pi. It is odd, so every an is 0 and only sine terms survive. Its first sine coefficient works out to b1 equals 4 over pi. The sawtooth f of x equals x behaves the same way: odd, sines only, and b2 equals minus 1. In practice you stop after a few terms, and the partial sum already looks like the wave except near jumps.
This works because sines and cosines of different frequencies are orthogonal: multiply two different ones and integrate over a full period, and you get zero. That is what isolates one coefficient at a time instead of solving for all of them together. Symmetry reads off whole families for free: odd functions keep sines only, and even functions keep cosines only.
Expect trouble exactly at jump discontinuities: the series struggles there and the partial sums overshoot near the jump. Engineers still love the split, because solving a system driven by a messy periodic input reduces to solving it one frequency at a time and adding the answers.
Multiply by each tone, integrate over a period, and stack the surviving sines and cosines.
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