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- https://math.libretexts.org/Bookshelves/Differential_Equations/A_Second_Course_in_Ordinary_Differential_Equations%3A_Dynamical_Systems_and_Boundary_Value_Problems_(Herman)/05%3A_Fourier_Series/5.02%3A_Fourier_Trigonometric_Series\[\dfrac{a_{0}}{2} \int_{0}^{2 \pi} \cos m x d x+\sum_{n=1}^{\infty}\left[a_{n} \int_{0}^{2 \pi} \cos n x \cos m x d x+b_{n} \int_{0}^{2 \pi} \sin n x \cos m x d x\right]. \label{5.6} \] \int_{0}^{2 \...\[\dfrac{a_{0}}{2} \int_{0}^{2 \pi} \cos m x d x+\sum_{n=1}^{\infty}\left[a_{n} \int_{0}^{2 \pi} \cos n x \cos m x d x+b_{n} \int_{0}^{2 \pi} \sin n x \cos m x d x\right]. \label{5.6} \] \int_{0}^{2 \pi} \cos n x \cos m x d x &=\dfrac{1}{2} \int_{0}^{2 \pi}[\cos (m+n) x+\cos (m-n) x] d x \\[4pt] \[\int_{0}^{2 \pi} \sin m x \cos m x d x=\dfrac{1}{2} \int_{0}^{2 \pi} \sin 2 m x d x=\dfrac{1}{2}\left[\dfrac{-\cos 2 m x}{2 m}\right]_{0}^{2 \pi}=0. \nonumber \]
- https://math.libretexts.org/Bookshelves/Differential_Equations/Partial_Differential_Equations_(Walet)/04%3A_Fourier_Series/4.05%3A_When_is_it_a_Fourier_SeriesThe Fourier coefficients are \[\begin{aligned} a_0 &= \frac{1}{5} \int_{-5}^0 -3 dx + \frac{1}{5} \int^{5}_0 3 dx = 0 \nonumber\\ a_n &= \frac{1}{5} \int_{-5}^0 -3 \cos\left(\frac{n\pi x}{5}\right) +\...The Fourier coefficients are \[\begin{aligned} a_0 &= \frac{1}{5} \int_{-5}^0 -3 dx + \frac{1}{5} \int^{5}_0 3 dx = 0 \nonumber\\ a_n &= \frac{1}{5} \int_{-5}^0 -3 \cos\left(\frac{n\pi x}{5}\right) +\frac{1}{5} \int_0^5 3 \cos\left(\frac{n\pi x}{5}\right) = 0\\ b_n &= \frac{1}{5} \int_{-5}^0 -3 \sin\left(\frac{n\pi x}{5}\right) +\frac{1}{5} \int_0^5 3 \sin\left(\frac{n\pi x}{5}\right) \nonumber\\ &= \left.\frac{3}{n\pi}\cos\left(\frac{n\pi x}{5}\right)\right|^0_{-5} -\left.\frac{3}{n\pi}\cos\le…
- https://math.libretexts.org/Bookshelves/Differential_Equations/Introduction_to_Partial_Differential_Equations_(Herman)/03%3A_Trigonometric_Fourier_Series/3.02%3A_Fourier_Trigonometric_SeriesOur goal is to find the Fourier series representation given f(x) . Having found the Fourier series representation, we will be interested in determining when the Fourier series converges and to what f...Our goal is to find the Fourier series representation given f(x) . Having found the Fourier series representation, we will be interested in determining when the Fourier series converges and to what function it converges.
- https://math.libretexts.org/Bookshelves/Differential_Equations/Differential_Equations_(Chasnov)/09%3A_Partial_Differential_Equations/9.03%3A_Fourier_SeriesThe orthogonality relations for \(n\) and \(m\) positive integers are then given with compact notation as the integration formulas \[\label{eq:2} \int_{-L}^L\cos\left(\frac{m\pi x}{L}\right)\cos\left(...The orthogonality relations for \(n\) and \(m\) positive integers are then given with compact notation as the integration formulas \[\label{eq:2} \int_{-L}^L\cos\left(\frac{m\pi x}{L}\right)\cos\left(\frac{n\pi x}{L}\right)dx=L\delta_{nm},\] \[\label{eq:3}\int_{-L}^L\sin\left(\frac{m\pi x}{L}\right)\sin\left(\frac{n\pi x}{L}\right)dx=L\delta_{nm},\] \[\label{eq:4}\int_{-L}^L\cos\left(\frac{m\pi x}{L}\right)\sin\left(\frac{n\pi x}{L}\right)dx=0.\]
