7.1 Periodic Functions and Trigonometric Series

A function \(f(x)\) is periodic if it is defined for all real \(x\) and there is a positive number \(p\) such that \[f(x + p) = f(x)\qquad \text {for all } x .\] The number \(p\) is a period of \(f\); the smallest such \(p\), if there is one, is the fundamental period. For example \(\sin x\) and \(\cos x\) have period \(2\pi \), and \(\sin nx\) and \(\cos nx\) have period \(\frac {2\pi }{n}\), hence also \(2\pi \).

The functions \[1,\quad \cos x,\quad \sin x,\quad \cos 2x,\quad \sin 2x,\quad \dots ,\quad \cos nx,\quad \sin nx,\quad \dots \] form the trigonometric system, and a series \begin {equation} a_0 + \sum ^{\infty }_{n=1}(a_n \cos nx + b_n \sin nx) = a_0 + a_1\cos x + b_1\sin x + a_2\cos 2x + b_2\sin 2x + \cdots \label {eq:trigseries} \end {equation} is a trigonometric series, with coefficients \(a_0\), \(a_n\) and \(b_n\). If it converges, its sum has period \(2\pi \).

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