Find the Fourier integral representation of f(x) = e −|x| and hence show that

Question

Question

(15 marks) Find the Fourier integral representation of $f(x)=e^{-|x|}$ and hence show that

$$ \int_{0}^{\infty} \frac{d t}{1+t^{2}}=\frac{\pi}{2} $$

Answer 1

Similar Homework Help Questions

Compute the Fourier cosine coefficients for f(x)

Let $f(x)= \begin{cases}0 & \text { for } 0 \leq x<2 \\ -(4-x) & \text { for } 2 \leq x \leq 4\end{cases}$- Compute the Fourier cosine coefficients for $f(x)$.- $a_{0}=$- $a_{n}=$- What are the values for the Fourier cosine series $\frac{a_{0}}{2}+\sum_{n=1}^{\infty} a_{n} \cos \left(\frac{n \pi}{4} x\right)$ at the given points.- $x=2:$- $x=-3$ :- $x=5:$
3. Find the Fourier sine integral representation for 3. Find the Fourier sine integral representation for

3. Find the Fourier sine integral representation for 3. Find the Fourier sine integral representation for
Determine the Fourier series representation of

Let $\left.x_{(} t\right)=\left\{\begin{array}{rr}t, & 0 \leq t \leq 1 \\ -t, & -1 \leq t \leq 0\end{array}\right.$, be a periodic signal with fundamental period of $T=2$ and Fourier series coefficients $a_{k}$.a) Sketch the waveform of $x(t)$ and $\frac{d x(t)}{d t}$ b) Calculate $a_{0}$ c) Determine the Fourier series representation of $g(t)=\frac{d x(t)}{d t}d) Using the results from Part (c) and the property of continuous-time Fourier series to determine the Fourier series coefficients of \(x(t)$

Find the Fourier transform of f(x) = 1–x?, for -1 < x < 1 and f(x)...

Find the Fourier transform of f(x) = 1–x?, for -1 < x < 1 and f(x) = 0 otherwise. Hence evaluate the integral 6 * * cos sin cos des.
Consider the heat conduction problem

5. If $f(x)=\left\{\begin{array}{cc}0 & -2<x<0 \\ x & 0<x<2\end{array} \quad\right.$is periodio of period 4 , and whose Fourier series is given by $\frac{a_{0}}{2}+\sum_{n=1}^{2}\left[a_{n} \cos \left(\frac{n \pi}{2} x\right)+b_{n} \sin \left(\frac{n \pi}{2} x\right)\right], \quad$ find $a_{n}$A. $\frac{2}{n^{2} \pi^{2}}$B. $\frac{(-1)^{n}-1}{n^{2} \pi^{2}}$C. $\frac{4}{n^{2} \pi^{2}}$D. $\frac{2}{n \pi}$$\mathbf{E}_{1} \frac{2\left((-1)^{n}-1\right)}{n^{2} \pi^{2}}$F. $\frac{4}{n \pi}$6. Let $f(x)-2 x-l$ on $[0,2]$. The Fourier sine series for $f(x)$ is $\sum_{w}^{n} b_{n} \sin \left(\frac{n \pi}{2} x\right)$, What is $b, ?$A. $\frac{4}{3 \pi}$B. $\frac{2}{\pi}$C. $\frac{4}{\pi}$D. $\frac{-4}{3 \pi}$E. $\frac{-2}{\pi}$F. $\frac{-4}{\pi}$7. Let $f(x)$ be periodic...
Check the existence of the Laplace transform for the given function and hence show that -...

Check the existence of the Laplace transform for the given function and hence show that - cos 20 1s² + 4 L = In t s2 where L{f(t)} is represent the Laplace transform of f(t). [Hint: 2 cos A cos B = COSIA+B) + cos(A - B) sin(A + B) + sin(A - B) = sinA cosB, sin(A + B) – sin(A - ?) = os AsmB] [2+ Find the Fourier Sine series of [8 f(x) = e-*,0<x<. Using the...

Use a power series to approximate the definite integral,

17 Use a power series to approximate the definite integral, $I$, to six decimal places.$$ \int_{0}^{0.4} \frac{x^{5}}{1+x^{7}} d x $$Find the radius of convergence, $R$, of the series.$$ \sum_{n=1}^{\infty} \frac{x^{n+4}}{4 n !} $$$$ R= $$Find the interval, $I$, of convergence of the series. (Enter your answer using interval notation.) $I=$
Math assignment about extreme of function and mean value theorem due tomorrow

The math assignment was due tomorrow so I need the answers with steps as soon as possible and I need it around today so please hurry1. (15 marks) Find the domain of $f(x)=2 x-\frac{1}{x^{2}}$ and find the intervals on which the function is increasing or decreasing.2. (15 marks) For $f(x)=\frac{x^{3}}{x+1}$, find the local maximum and minimum.3. (15 marks) Find the absolute maximum and minimum of the function $f(x)=\frac{x}{1+x^{2}}$ on the closed interval $[-1,2]$.4. (20 marks) Sketch the graph of the...
4. Compute Fourier integral representation for f(x) = S\sinx[xST [x] > T = {leine $ "...

4. Compute Fourier integral representation for f(x) = S\sinx[xST [x] > T = {leine $ " cosas # 1 (cos") ds = TT and deduce that 2:

Find the periodic solutions of the differential equations

Find the periodic solutions of the differential equations $(a) \frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{ky}=\mathrm{f}(\mathrm{x}),(b) \frac{\mathrm{d}^{3} \mathrm{y}}{\mathrm{d}^{3} \mathrm{x}}+\mathrm{ky}=\mathrm{f}(\mathrm{x})$where $k$ is a constant and $f(x)$ is a $2 \pi$ - periodic function.Consider a Fourier series expansion for $f(x)$ using the complex form, $f(x)=\sum_{n=-\infty}^{n=+\infty} f_{n} e^{i n x}$ and try a solution of the form $y(x)=\sum_{n=-\infty}^{n=+\infty} y_{n} e^{i n x}$