7.2 Orthogonality of the Trigonometric System

The trigonometric system is orthogonal on the interval \(-\pi \leq x \leq \pi \) (hence on any interval of length \(2\pi \)). Hence, for any integer \(m\) and \(n\), we get \[* (1)\hspace {1cm} \int ^{\pi }_{-\pi } \cos m x \cos n x dx = 0\hspace {0.4cm} ( m\neq n) \]

\[m = n \implies \int ^{\pi }_{-\pi } \cos ^2 mx\,dx = \frac {1}{2}\int ^{\pi }_{-\pi }\big ( 1 + \cos 2mx\big )dx = \pi \]

\[\int ^{\pi }_{-\pi }\sin mx \sin nx dx = 0 \hspace {0.4cm} (m\neq n) \hspace {0.3cm} \neq 0\hspace {0.3cm}\text {if}\hspace {0.2cm} m = n\]

\[\int ^{\pi }_{-\pi }\cos mx \sin nx\,dx = 0 \hspace {0.4cm} \forall m,n = 0\hspace {0.5cm} (\text {includes} \hspace {0.3cm} m = n)\]

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