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Problem 398

Consider the infinite trigonometric series $\left(a_{0} / 2\right)+{ }^{\infty} \Sigma_{n=1}\left(a_{n} \cos n x+b_{n} \sin n x\right)$ and assume that it converges uniformly for all \(\mathrm{x} \in(-\pi, \pi)\). It can then be considered as a function \(f\) of \(x\) with period \(2 \pi\), i.e. $\mathrm{f}(\mathrm{x})=\left(\mathrm{a}_{0} / 2\right)+{ }^{\infty} \sum_{\mathrm{n}=1} \mathrm{a}_{\mathrm{n}} \cos \mathrm{n} \mathrm{x}+\mathrm{b}_{\mathrm{n}} \sin \mathrm{n} \mathrm{x}$ Determine the values of \(a_{n}, b_{n}\) in terms of \(f(x)\)

Expert verified

In summary, the values of \(a_n\) and \(b_n\) in terms of \(f(x)\) can be determined using the following formulas:
\(a_0 = \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x) dx\)
\(a_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x) \cos(nx) dx\)
\(b_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x) \sin(nx) dx\)
These expressions can be used to find the Fourier series coefficients for any given periodic function \(f(x)\) with period \(2\pi\).

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Chapter 14

Find the Fourier cosine series over the interval \(0<\mathrm{x}<\mathrm{c}\) for the function \(\mathrm{f}(\mathrm{x})=\mathrm{x}\)

Chapter 14

Find the Fourier sine series for the function defined by
$$
\begin{array}{ll}
f(x)=0 & 0 \leq x<\pi / 2 \\
f(x)=1 & \pi / 2

Chapter 14

Define the following properties of a real valued function \(\mathrm{f}\) of a real variable: (a) The "limit from the right" of \(\mathrm{f}\) at \(\mathrm{x}_{\mathrm{o}}\) (b) The "limit from the left" of \(\mathrm{f}\) at \(\mathrm{x}_{\circ}\) (c) \(\mathrm{f}\) is piecewise continuous on \((\mathrm{a}, \mathrm{b})\). (d) The right and left hand derivatives of \(\mathrm{f}\) at \(\mathrm{x}_{0}\)

Chapter 14

The Fourier series of the following functions have been found in previous
problems but no convergence questions were discussed. Determine now which of
these functions has Fourier series which are
(i) pointwise convergent
(ii) uniformly convergent and
(iii) convergent in the mean:
(a) $\mathrm{f}(\mathrm{x})=\mathrm{e}^{\mathrm{x}}
\quad(-\pi<\mathrm{x}<\pi)$
(b) \(\mathrm{f}(\mathrm{x})=\mathrm{x} \quad, 0<\mathrm{x}<\pi\)
and \(=0 \quad, \pi

Chapter 14

Let \(\mathrm{f}(\mathrm{x})\) be a real valued function of one variable, \(\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}\), with a period \(2 c\) (i.e. \(\mathrm{f}(\mathrm{x}+2 \mathrm{c})=\mathrm{f}(\mathrm{x})\) for all \(\mathrm{x} \in \mathrm{R}\) ). Define the Fourier series of \(\mathrm{f}\).

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