7.1 Periodic Functions and Trigonometric Series

A function \(f(x)\) is called periodic if it is defined for all real \(x\) and if there is some positive number \(p\ni \) \[f(x + p) = f(x) \hspace {0.5cm} \forall x \hspace {0.5cm}\cdots \cdots \cdots \hspace {0.5cm} (a)\] The number \(p\) is called the period of \(f(x)\).

The set of functions \[-1\hspace {0.1cm}, \hspace {0.5cm} \sin x\hspace {0.1cm} , \hspace {0.5cm} \cos x\hspace {0.1cm}, \hspace {0.5cm} \cos 2x\hspace {0.1cm},\hspace {0.5cm} \sin 2x\hspace {0.1cm}, \cdots \cdots \cdots \hspace {0.1cm} \cos nx\hspace {0.1cm},\hspace {0.5cm} \sin nx, \hspace {0.5cm} n = 1,2,\cdots \cdots \hspace {0.5cm} (b)\] is called the trigonometric system and the series \[a_0 + a_1\cos x + b_1\sin x + a_2\cos 2x + b_2\sin 2x + \cdots \cdots + a_n\cos nx + b_n \sin nx + \cdots \cdots \] \[\cdots \cdots =\sum ^{\infty }_{n=1}\big (a_n \cos nx + b_n \sin nx\big ) + a_0 \hspace {0.5cm} \cdots \cdots \cdots \hspace {0.3cm} (c)\] is called Trigonometric series, and the numbers \(a_0, a_n\) and \(b_n\hspace {0.4cm} (n = 1,2,\cdots \cdots )\) are coefficients of the series.

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