5.8 The Complex Form
Writing the series with exponentials rather than sines and cosines shortens every formula and is the form in which the subject is usually met beyond this chapter.
Theorem 5.8.1 (Complex Fourier series). With \(c_n = \tfrac 12\left (a_n - i b_n\right )\) for \(n>0\), \(c_{-n} = \overline {c_n}\) and \(c_0 = a_0/2\), \[f(x) = \sum ^{\infty }_{n=-\infty }c_n e^{inx}, \qquad c_n = \frac {1}{2\pi }\int ^{\pi }_{-\pi }f(x)\,e^{-inx}\,dx .\]
Proof. Substitute \(\cos nx = \tfrac 12\left (e^{inx}+e^{-inx}\right )\) and \(\sin nx = \tfrac {1}{2i}\left (e^{inx}-e^{-inx}\right )\) into the real series and collect the coefficient of \(e^{inx}\). The formula for \(c_n\) follows from the orthogonality relation \(\int ^{\pi }_{-\pi }e^{inx}e^{-imx}dx = 2\pi \,\delta _{nm}\). □
Note 5.8.2. One formula now replaces three, and the single orthogonality relation \(\int e^{inx}\overline {e^{imx}}dx = 2\pi \delta _{nm}\) replaces the separate relations for \(\cos \cos \), \(\sin \sin \) and \(\sin \cos \). In this form Parseval reads \[\frac {1}{2\pi }\int ^{\pi }_{-\pi }\left |f\right |^{2}dx = \sum ^{\infty }_{n=-\infty }\left |c_n\right |^{2},\] which is as clean a statement of the Pythagorean idea as one could want.
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