Mth3006 Weekly Problems 2

Question One

A function is defined by

$$T(x)=\{\begin{array}{ll}1& \text{when }|x|<a\\ 0& \text{when }|x|>a\end{array}$$

Where $a$ is a real, positive constant.

Part A

Sketch the graph of $T(x)$, and use your result to sketch the graph of the convolution

$$h(x)=T(x)\ast [\delta (x-b)+\delta (x+b)]$$

when $b>a$.

Solution

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Part B

Use the convolution theorem to calculate $\tilde{h}(k)$, the Fourier transform of$h(x)$, and show that $\tilde{h}(k)$ has zeros at $k=n\pi /a$ when $n\ne 0$ and $k=[(n+1/2)\pi ]/b$ for all $n\in \mathbb{Z}$.

Solution

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Question Two

Part A

Find the Fourier transform of the function $f(t)=\exp (-|t|)$. Before starting the calculation, think carefully about which form of the transform will give the most straightforward integral.

Solution

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Part B

Take the inverse Fourier transform of your result from part (a) to show that

$$\frac{\pi }{2}\exp (-|t|)={\int }_0^{\infty }\frac{\cos (\omega t)}{1+{\omega }^2}d\omega$$

Solution

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Question Three

Ignoring the point $t=0$, the Heaviside step function is defined by

$$H(t)=\{\begin{array}{ll}1& \text{when }t>0\\ 0& \text{when }t<0\end{array}$$

Part A

Calculate the Fourier transform of $f(t)=e^{-at}H(t)$, where $a>0$ is a real constant.

Solution

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Part B

Use your result, together with the convolution theorem, to evaluate the inverse Fourier transform of $\tilde{f}(\omega )=\frac{1}{2\pi (a+i\omega {)}^2}$

Hint: Write $\tilde{f}(\omega )$ in terms of your answer to part A.

Solution

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