Question

PS3 Q2. Consider the function $f(x) = e^x$. (i) Find the Fourier Sine Series (FSS) coefficients for $f$ on $[0, l] = [0, \pi]$. That is, find $b_n$ in $\infty$ $f(x) \sim \sum_{n=1} b_n \sin(\frac{n\pi x}{l})$ Plot your FSS on $[0, l] = [0, \pi]$ using 5 and 20 terms and superimpose the graph of $f$. (ii) Find the Fourier Cosine Series coefficients for $f$ on $[0, l]$. That is, find $a_n$ in $\infty$ $f(x) \sim a_0 + \sum_{n=1} a_n \cos(\frac{n\pi x}{l})$ Plot your FCS on $[0, l] = [0, \pi]$ using 5 and 20 terms and superimpose the graph of $f$. (iii) Find the Full Fourier Series (FS) coefficients for $f$ on $[-l, l] = [-\pi, \pi]$. That is find $a_n, b_n$ in $\infty$ $f(x) \sim a_0 + \sum_{n=1} (a_n \cos(\frac{n\pi x}{l}) + b_n \sin(\frac{n\pi x}{l}))$ Plot your FS on $[-l, l]$ using 5 and 20 terms and superimpose the graph of $f$. (iv) What can you say about the answers you obtained in Parts (i)-(iii). That is, are any of the coefficients you obtained in the different parts the same, and why? Also, discuss the convergence of the 3 Fourier Series on the intervals considered. Here, Hillen's Theorem 2.3 (Dirichlet's Theorem) should be applied for Parts (i)-(iii).

          PS3 Q2. Consider the function $f(x) = e^x$.
(i) Find the Fourier Sine Series (FSS) coefficients for $f$ on $[0, l] = [0, \pi]$. That is, find $b_n$ in
$\infty$
$f(x) \sim \sum_{n=1} b_n \sin(\frac{n\pi x}{l})$
Plot your FSS on $[0, l] = [0, \pi]$ using 5 and 20 terms and superimpose the graph of $f$.
(ii) Find the Fourier Cosine Series coefficients for $f$ on $[0, l]$. That is, find $a_n$ in
$\infty$
$f(x) \sim a_0 + \sum_{n=1} a_n \cos(\frac{n\pi x}{l})$
Plot your FCS on $[0, l] = [0, \pi]$ using 5 and 20 terms and superimpose the graph of $f$.
(iii) Find the Full Fourier Series (FS) coefficients for $f$ on $[-l, l] = [-\pi, \pi]$. That is find
$a_n, b_n$ in
$\infty$
$f(x) \sim a_0 + \sum_{n=1} (a_n \cos(\frac{n\pi x}{l}) + b_n \sin(\frac{n\pi x}{l}))$
Plot your FS on $[-l, l]$ using 5 and 20 terms and superimpose the graph of $f$.
(iv) What can you say about the answers you obtained in Parts (i)-(iii). That is, are any of
the coefficients you obtained in the different parts the same, and why? Also, discuss the
convergence of the 3 Fourier Series on the intervals considered. Here, Hillen's Theorem
2.3 (Dirichlet's Theorem) should be applied for Parts (i)-(iii).

PS3 Q2. Consider the function f(x) = e^x.
(i) Find the Fourier Sine Series (FSS) coefficients for f on [0, l] = [0, π]. That is, find bn in
∞
f(x) ∼∑n=1 bn sin((nπ x)/(l))
Plot your FSS on [0, l] = [0, π] using 5 and 20 terms and superimpose the graph of f.
(ii) Find the Fourier Cosine Series coefficients for f on [0, l]. That is, find an in
∞
f(x) ∼ a0 + ∑n=1 an cos((nπ x)/(l))
Plot your FCS on [0, l] = [0, π] using 5 and 20 terms and superimpose the graph of f.
(iii) Find the Full Fourier Series (FS) coefficients for f on [-l, l] = [-π, π]. That is find
an, bn in
∞
f(x) ∼ a0 + ∑n=1 (an cos((nπ x)/(l)) + bn sin((nπ x)/(l)))
Plot your FS on [-l, l] using 5 and 20 terms and superimpose the graph of f.
(iv) What can you say about the answers you obtained in Parts (i)-(iii). That is, are any of
the coefficients you obtained in the different parts the same, and why? Also, discuss the
convergence of the 3 Fourier Series on the intervals considered. Here, Hillen's Theorem
2.3 (Dirichlet's Theorem) should be applied for Parts (i)-(iii).

Added by Melinda R.

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