The trigonometric series

Writing a periodic function as an infinite sum of sines and cosines, with coefficients extracted by orthogonality integrals taken over one period.

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4 exercises

Suppose $f(t)$ is defined on $[-\pi,\pi]$ as $f(t) = \sin(t)$ . Extend periodically and compute the Fourier series.

Suppose $f(t)$ is defined on $(-\pi,\pi]$ as $f(t) = \sin(\pi t)$ . Extend periodically and compute the Fourier series.

Suppose $f(t)$ is defined on $(-\pi,\pi]$ as $f(t) = \sin^2(t)$ . Extend periodically and compute the Fourier series.

Suppose $f(t)$ is defined on $(-\pi,\pi]$ as $f(t) = t^4$ . Extend periodically and compute the Fourier series.