Fourier coefficients
Fourier coefficients: read them off a trigonometric polynomial, expand products and powers of sines and cosines, compute them by the formulas.
How to solve it
A Fourier series breaks a function into sines and cosines of different frequencies. The key is orthogonality: the integral of the product of two different harmonics over a period is zero. So each coefficient is picked out by its own integral, and for a trigonometric polynomial it is simply visible.
Step by step
- If the function is already a sum of sines and cosines, read the coefficients off it. The constant term is $\frac{a_0}{2}$.
- Expand products and powers into sums: $\cos^2 x = \frac{1 + \cos 2x}{2}$, $\sin 2x\cos x = \frac{\sin 3x + \sin x}{2}$, $\cos^3 x = \frac{3\cos x + \cos 3x}{4}$.
- An odd function has only sines ($a_n = 0$), an even one only cosines ($b_n = 0$). The integral of an odd function over a symmetric interval is zero.
- Otherwise use the formulas, integrating by parts.
Common mistakes
- Forgetting that the constant term is half of $a_0$: if $f = 5 + \ldots$, then $a_0 = 10$.
- Reading a coefficient off a product without expanding it: $\cos^3 x$ does not mean “$a_3 = 1$”.
- Ignoring parity and computing needless integrals — or losing the sign of an odd function.
- Forgetting the factor $\frac{1}{\pi}$.