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Chapter 8: Problem 5

Find a Fourier series representation of the Dirac delta function, \(\delta(x)\), on \([-L, L]\)

Short Answer

Expert verified
The Fourier series representation of the Dirac Delta function on the interval \([-L, L]\) is \(\delta(x) \approx 1/2L\).

Step by step solution

01

Understand the Dirac Delta Function

The Dirac Delta function, denoted as \(\delta(x)\), is a function that is zero everywhere except for the point at 0, where it is undefined, and its integral over the entire real line is equal to 1. It is often considered as 'sampling' function.
02

Establish Fourier Transform Pair for Dirac Delta Function

We have the general representation for a Fourier transform pair as follows: Fourier transform of \(\delta(x)\) is given by \(F(w) = \int_{-\infty}^{+\infty} \delta(x) e^{-iwx} dx\). We know that integral of \(\delta(x)\) over \(-∞\) to \(+∞\) equals 1. So, Fourier transform of \(\delta(x)\) equals 1 for all w, that is \(F(w) = 1\). The inverse Fourier transform of 1 is given by \(ƒ(x) = \int_{-\infty}^{+\infty} 1 * e^{iwx} dw\), which yields the dirac delta function. Thus the Fourier transform pair is established, \(\delta(x) \longleftrightarrow 1\).
03

Find Fourier Series Representation

Having established the Fourier transform pair, we can next move to finding the Fourier Series representation. The representation for any function in Fourier series within an interval \([-L, L]\) is given by \(ƒ(x) ≈ (1/2L) \int_{-L}^{L} dw e^{iwx}\), substituting \(\delta(x)\) in this we have \(\delta(x) \approx (1/2L) \int_{-L}^{L} dw e^{iwx}\).
04

Simplify the Fourier Series Representation

The integral of \(e^{iwx}\) runs from \(-L\) to \(L\), this implies that this is a periodic function and its average value over one period is constant as 1. Therefore, Fourier series of \(\delta(x)\) over \([-L, L]\) simplifies to \(\delta(x) \approx 1/2L\).

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Most popular questions from this chapter

Find the inverse Laplace transform of the following functions using the properties of Laplace transforms and the table of Laplace transform pairs. a. \(F(s)=\frac{18}{s^{3}}+\frac{7}{s} .\) b. \(F(s)=\frac{1}{s-5}-\frac{2}{s^{2}+4}\). c. \(F(s)=\frac{s+1}{s^{2}+1} .\) d. \(F(s)=\frac{3}{s^{2}+2 s+2}\) e. \(F(s)=\frac{1}{(s-1)^{2}}\) f. \(F(s)=\frac{e^{-3 s}}{s^{2}-1} .\) g. \(F(s)=\frac{1}{s^{2}+4 s-5}\) h. \(F(s)=\frac{s+3}{s^{2}+8 s+17} .\)

Use the result from Problem 6 plus properties of the Fourier transform to find the Fourier transform, of \(f(x)=x^{2} e^{-a|x|}\) for \(a>0\).

A damped harmonic oscillator is given by $$ f(t)=\left\\{\begin{array}{cl} A e^{-a t} e^{i \omega_{0} t}, & t \geq 0 \\ 0, & t<0 \end{array}\right. $$ a. Find \(\hat{f}(\omega)\) and b. the frequency distribution \(|\hat{f}(\omega)|^{2}\). c. Sketch the frequency distribution.

Given the following Laplace transforms \(F(s)\), find the function \(f(t)\). Note that in each case there are an infinite number of poles, resulting in an infinite series representation. a. \(F(s)=\frac{1}{s^{2}\left(1+e^{-5}\right)}\) b. \(F(s)=\frac{1}{s \sinh s}\). c. \(F(s)=\frac{\text { sinh } s}{s^{2} \cosh s}\)

For the case that a function has multiple roots, \(f\left(x_{i}\right)=0, i=1,2, \ldots\), it can be shown that $$ \delta(f(x))=\sum_{i=1}^{n} \frac{\delta\left(x-x_{i}\right)}{\left|f^{\prime}\left(x_{i}\right)\right|} $$ Use this result to evaluate \(\int_{-\infty}^{\infty} \delta\left(x^{2}-5 x-6\right)\left(3 x^{2}-7 x+2\right) d x\).
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