[CHWiSGevordTech] Oefeningen H9 Fourier -- Sage

Oef.3a

f=piecewise([((-pi,0),0),((0,pi),x)]) plot(f,-pi,pi)

fR=f.fourier_series_partial_sum(6,pi) fR

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{4} \, \pi - \frac{2 \, \cos\left(5 \, x\right)}{25 \, \pi} - \frac{2 \, \cos\left(3 \, x\right)}{9 \, \pi} - \frac{2 \, \cos\left(x\right)}{\pi} + \frac{1}{5} \, \sin\left(5 \, x\right) - \frac{1}{4} \, \sin\left(4 \, x\right) + \frac{1}{3} \, \sin\left(3 \, x\right) - \frac{1}{2} \, \sin\left(2 \, x\right) + \sin\left(x\right)$

var('n') assume(n,'integer') an=f.fourier_series_cosine_coefficient(n,pi) an

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{\left(-1\right)^{n} - 1}{\pi n^{2}}$

assume(n,'even') an=f.fourier_series_cosine_coefficient(n,pi) an

$\newcommand{\Bold}[1]{\mathbf{#1}}0$

forget() assume(n,'odd') an=f.fourier_series_cosine_coefficient(n,pi) an

$\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{2}{\pi n^{2}}$

forget() assume(n,'integer') bn=f.fourier_series_sine_coefficient(n,pi) bn

$\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{\left(-1\right)^{n}}{n}$

forget()

plot(f,-pi,pi)+plot(fR,-pi,pi,color='red')

fu=piecewise([((-2*pi,-pi),x+2*pi),((-pi,0),0),((0,pi),x),((pi,2*pi),0)]) plot(fu,-2*pi,2*pi)+plot(fR,-2*pi,2*pi,color='red')

Oef 4

Gegeven de functie met periode $2\pi$:
\begin{equation*}
f(x) =
\begin{cases}
\pi & \text{als}\ -\pi\leq x \leq 0 \\
\pi-2x & \text{als}\ 0 < x\leq\pi.
\end{cases}
\end{equation*}

(a) Bepaal het even en oneven deel van de functie $f(x)$. Tip: maak gebruik van het feit dat $f(x) = f_e(x) + f_o(x)$ met $f_e(x) = (f(x) + f(-x))/2$ en $f_o(x) = (f(x) - f(-x))/2$.

Maak een figuur waarop de drie functies ($f$ en haar even deel $f_e$ en oneven deel $f_o$ ) voorgesteld worden en controleer op de figuur dat $f(x) = f_e(x) + f_o(x)$.

f=piecewise([((-pi,0),pi),((0,pi),pi-2*x)]) fe=piecewise([((-pi,0),pi+x),((0,pi),pi-x)]) fo=piecewise([((-pi,0),-x),((0,pi),-x)]) h=(fe+fo) plot(f,-pi,pi)+plot(fe,(-pi,pi),color='green')+plot(fo,(-pi,pi),color='red')+plot(h,(-pi,pi),color='yellow',linestyle='--')

(b) Bepaal de Fourier-reeksontwikkeling van $f_e(x)$, van $f_o(x)$ en van $f(x)$ tot en met de termen in $cos(5x)$ en $sin(5x)$. Wat stel je vast?

feN=fe.fourier_series_partial_sum(6,pi) feN

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2} \, \pi + \frac{4 \, \cos\left(5 \, x\right)}{25 \, \pi} + \frac{4 \, \cos\left(3 \, x\right)}{9 \, \pi} + \frac{4 \, \cos\left(x\right)}{\pi}$

foN=fo.fourier_series_partial_sum(6,pi) foN

$\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{2}{5} \, \sin\left(5 \, x\right) + \frac{1}{2} \, \sin\left(4 \, x\right) - \frac{2}{3} \, \sin\left(3 \, x\right) + \sin\left(2 \, x\right) - 2 \, \sin\left(x\right)$

fN=f.fourier_series_partial_sum(6,pi) fN

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2} \, \pi + \frac{4 \, \cos\left(5 \, x\right)}{25 \, \pi} + \frac{4 \, \cos\left(3 \, x\right)}{9 \, \pi} + \frac{4 \, \cos\left(x\right)}{\pi} - \frac{2}{5} \, \sin\left(5 \, x\right) + \frac{1}{2} \, \sin\left(4 \, x\right) - \frac{2}{3} \, \sin\left(3 \, x\right) + \sin\left(2 \, x\right) - 2 \, \sin\left(x\right)$

f=piecewise([((-pi,0),pi),((0,pi),pi-2*x),((pi,2*pi),pi),((2*pi,3*pi),5*pi-2*x)]) plot(f,-pi,3*pi)+plot(fN,(-pi,3*pi),color='red')

a0=(1/2)*f.fourier_series_cosine_coefficient(0,pi) a0

$\newcommand{\Bold}[1]{\mathbf{#1}}\pi$

var('n') an=fe.fourier_series_cosine_coefficient(n,pi) an

$\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{2 \, {\left(\cos\left(\pi n\right) - 1\right)}}{\pi n^{2}}$

assume(n,'integer') an=fe.fourier_series_cosine_coefficient(n,pi) an

$\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{2 \, {\left(\left(-1\right)^{n} - 1\right)}}{\pi n^{2}}$

assume(n,'even') an=fe.fourier_series_cosine_coefficient(n,pi) an

$\newcommand{\Bold}[1]{\mathbf{#1}}0$

forget() assume(n,'odd') an=fe.fourier_series_cosine_coefficient(n,pi) an

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{4}{\pi n^{2}}$

forget() bn=fo.fourier_series_sine_coefficient(n,pi) bn

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{2 \, {\left(\pi n \cos\left(\pi n\right) - \sin\left(\pi n\right)\right)}}{\pi n^{2}}$

forget() assume(n,'integer') bn=fo.fourier_series_sine_coefficient(n,pi) bn

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{2 \, \left(-1\right)^{n}}{n}$

assume(n,'even') bn=fo.fourier_series_sine_coefficient(n,pi) bn

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{2}{n}$

forget() assume(n,'odd') bn=fo.fourier_series_sine_coefficient(n,pi) bn

$\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{2}{n}$

forget()

Besluit:

$a_0= \frac{\pi}{2}$

$a_n = \begin{cases} 0 & \text{ als } n \text{ even is }\\ \frac{4}{\pi n^2} & \text{ als } n \text{ oneven is } \end{cases}$

$b_n= \frac{2(-1)^n}{n} = \begin{cases}\frac{2}{n} & \text{ als } n \text{ even is }\\ \frac{-2}{n}& \text{ als } n \text{ oneven is } \end{cases}$

[CHWiSGevordTech] Oefeningen H9 Fourier

1455 days ago by Carmen.Streat

Oef.3a

Oef 4