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Chapter 12: Problem 10

Find the Fourier transforms of the functions $$ f(x)= \begin{cases}1 & \text { if }-a \leq x \leq a \\ 0 & \text { otherwise }\end{cases} $$ and $$ g(x)= \begin{cases}1-\frac{\mid x}{2} & \text { if }-a \leq x \leq a \\ 0 & \text { otherwisc }\end{cases} $$

Short Answer

Expert verified
The Fourier transforms of \(f(x)\) and \(g(x)\) will be complex exponential functions of \(k\), obtained by computing their respective integrals on the interval \(-a \leq x \leq a\). These integrals will need to be solved using standard rules and methods of integration.

Step by step solution

01

Compute the Fourier Transform of Function f(x)

The Fourier transform of a function \(f(x)\) is given by the formula: \[ F(k) = \int^{+\infty}_{-\infty} f(x) \; e^{-2\pi ikx} \; dx \] But our function is a piecewise function that is only non-zero in the interval \(-a \leq x \leq a\). So, the computation is: \\[ F_f(k) = \int^{+a}_{-a} e^{-2\pi ikx} \; dx\]Which can be solved using standard methods of integration.
02

Compute the Fourier Transform of Function g(x)

Next, compute the Fourier transform of function \(g(x)\). Again, this function is a piecewise function that is only non-zero in the interval \(-a \leq x \leq a\). So, the computation becomes:\[ F_g(k) = \int^{+a}_{-a} (1-\frac{|x|}{2}) e^{-2\pi ikx} \; dx\]This integral is more complex due to the \(|x|\) part of the function. It would need to be split into two parts and solved accordingly, one for the range \(-a \leq x \leq 0\) and the other for the range \(0 \leq x \leq a\).
03

Evaluate the Results

Finally, evaluate the integrals to obtain the Fourier transforms of the functions f(x) and g(x). Solve the integrals using standard integration rules and methods, which should result in complex exponential functions of \(k\), which denote the frequency domain representation of our original functions.

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Most popular questions from this chapter

Show that the Green's function for the time-independent Klein-Gordon equation $$ \left(\nabla^{2}-m^{2}\right) \phi=\rho(r) $$ can be expressed as the Fourier integral $$ G\left(x-x^{\prime}\right)=-\frac{1}{(2 \pi)^{3}} \iiint d^{3} k \frac{e^{u k\left(x-y^{\prime}\right)}}{k^{2}+m^{2}} $$ Evaluate this integral and show that it results in $$ G(\mathbf{R})=-\frac{\mathrm{e}^{-\operatorname{m} k}}{4 \pi R} \quad \text { where } \quad \mathbf{R}=\mathbf{x}-\mathbf{x}^{\prime}, \quad R=|\mathbf{R}| $$ Find the solution \(\phi\) correspondmg to a point source $$ \rho(\mathbf{r})=q \delta^{3}(r) $$

Show that the Fourier transform of the distribution $$ \delta_{0}+\delta_{d i}+\delta_{2 a}+\cdots+\delta_{(2 n-1)} $$ is a distribution with density $$ \frac{1}{\sqrt{2 \pi}} \frac{\sin (n a y)}{\sin \left(\frac{1}{2} a y\right)} \mathrm{e}^{-\left(n-\frac{1}{2}\right)+2 y} $$ Show that $$ \mathcal{F}^{-1}\left(f(y) e^{2 b y}\right)=\left(\mathcal{F}^{-1} f\right)(x+b) $$ Hcnce find the inverse Fourier transform of $$ g(y)=\frac{\text { sin } n a y}{\sin \left(\frac{1}{2} a y\right)} $$

Which of the following is a distribution? (a) \(T(\phi)=\sum_{n=1}^{m} \lambda_{n} \phi^{(n)}(0) \quad\left(\lambda_{n} \in \mathbb{R}\right)\) (b) \(T(\phi)=\sum_{n=1}^{m} \lambda_{n} \phi\left(x_{n}\right) \quad\left(\lambda_{n}, x_{n} \in \mathbb{R}\right)\). (c) \(T(\phi)=(\phi(0))^{2}\). (d) \(T(\phi)=\sup \phi\) (c) \(T(\phi)=\int_{-\infty}^{\infty}|\phi(x)| \mathrm{d} x\).

Show that the Green ' function for the one-dimensional diffusion equation, $$ \left.\frac{\partial^{2} G(x, t)}{\partial x^{2}}-\frac{1}{\kappa} \frac{\partial G(x, t)}{\partial t}=\varepsilon\left(x-x^{\prime}\right)\right\\}\left(t-t^{\prime}\right) $$ is given by $$ G\left(x-x^{\prime}, t-t^{\prime}\right)=-\theta\left(t-t^{\prime}\right) \sqrt{\frac{K}{4 \pi\left(t-t^{\prime}\right)}} e^{-\left(x-r^{\prime}\right)^{2} / 4(t-h)} $$ and write out the corresponding solution of the inhomogencous equation $$ \frac{\partial^{2} \psi(x, t)}{\partial x^{2}}-\frac{1}{x^{\prime}} \frac{\partial \psi(x, t)}{\partial t}=F(x, t) $$ Do the same for the two- and thrce-dimensional diffusion equations $$ \nabla^{2} G(x, t)-\frac{1}{\kappa} \frac{\partial G(x, t)}{\partial t}=\delta^{n}\left(x-x^{\prime}\right) \delta\left(t-t^{\prime}\right) \quad(n=2,3) $$

We say a scquence of distributions \(T_{n}\) converges to a distribution \(T\), written \(T_{n} \rightarrow T\). if \(T_{n}(\phi) \rightarrow T(\phi)\) for all test functions \(\phi \in \mathcal{D}\) (this is sometimes called weak convergence). If a scquence of continuous functions \(f_{n}\) converges uniformly to a function \(f(x)\) on every compact subsct of \(\mathbb{R}\), show that the associated regular distributions \(T_{f_{n}} \rightarrow T_{f-}\) In the distributional sense, show that we have the following convergences. $$ \begin{aligned} f_{n}(x) &=\frac{n}{\pi\left(1+n^{2} x^{2}\right)} \rightarrow \delta(x) \\ g_{n}(x) &=\frac{n}{\sqrt{\pi}} \mathrm{e}^{-\alpha^{2} x^{2}} \rightarrow \delta(x) \end{aligned} $$
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