All courses › Laplace, Fourier and PDEs › Fourier coefficients

Fourier coefficients

A periodic function can be written as a sum of sine and cosine terms, a Fourier series. The coefficients $a_n$ and $b_n$ are found by integrating the function against cosine and sine respectively.

a_n = \frac1L\int_{-L}^{L}f(x)\cos\frac{n\pi x}{L}\,dx

cosine coefficients

b_n = \frac1L\int_{-L}^{L}f(x)\sin\frac{n\pi x}{L}\,dx

sine coefficients

Symbols

$2L$	period
$a_0/2$	mean value of f

Example

$f(x)=5+4\cos x$ on $[-\pi,\pi]$ : the mean value is $a_0/2=5$ , because the cosine term has mean value 0.

The integrand with sine gives

b_n

, and the one with cosine gives

a_n

- easy to mix up.

Practise fourier series for free →

← Shift rule and partial fractions · Odd and even functions →

Part of Laplace, Fourier and PDEs: Fourier series.