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Odd and even functions

The symmetry of a function greatly simplifies its Fourier series: an odd function has only sine terms, and an even function has only cosine terms (and a constant term).

\text{odd } f:\ a_n=0\ (\text{sine terms only})

odd function

\text{even } f:\ b_n=0\ (\text{cosine terms only})

even function

Symbols

$a_n$	cosine coefficients
$b_n$	sine coefficients

Example

$f(x)=x$ on $(-\pi,\pi)$ is odd: $b_n=\dfrac{2(-1)^{n+1}}{n}$ , so $b_5=\dfrac{2}{5}=0.4$ .

\sin

is odd and

\cos

is even - that is why the symmetry of

f

decides which terms vanish.

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Part of Laplace, Fourier and PDEs: Fourier series.