Mth3006 Weekly Problems 1

Question One

Calculate the Fourier transform of the delta function $\delta (t-a)$.

Solution

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Question Two

Calculate the inverse Fourier transform of

$$\tilde{f}(\omega )=\delta (\omega -{\omega }_0)-\delta (\omega +{\omega }_0)$$

, where ${\omega }_0$ is a real, non-zero constant, writing your answer in terms of a trigonometric function.

Solution

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Question Three

Find the Fourier transform of

$$f(t)=\{\begin{array}{ll}-1& \text{when }-1<t<0\\ 1& \text{when }0<t<1\\ 0& \text{otherwise}\end{array}$$

First by using the exponential form of the transform and then by using

$$\tilde{f}(\omega )=-i\sqrt{\frac{2}{\pi }}{\int }_0^{\infty }f(t)\sin (\omega t)dt$$

(valid since the function is odd). The two results should be the same.

Solution

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Question Four

Use an appropriate form of the Fourier transform to calculate

$$\tilde{f}(\omega )$$

when

$$f(t)=\{\begin{array}{ll}1-t^2& \text{when }|t|<1\\ 0& \text{when }|t|\ge 1\end{array}$$

And use your result to show that

$${\int }_0^{\infty }\frac{\sin x-x\cos x}{x^3}\cos \left(\frac{x}{2}\right)dx=\frac{3\pi }{16}$$

Solution

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Question Five

Find the Fourier cosine transform of $f(t)=\exp (-m|t|)$, $m>0$, and use your result to show that

$${\int }_0^{\infty }\frac{\cos (pv)}{v^2+{\beta }^2}dv=\frac{\pi }{2\beta }\exp (-p\beta )$$

Solution

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