10.9: Exercises

Exercise \(\PageIndex{1}\)

Find the relationship between the coefficients \(\{\alpha_n, \beta_m\}\) in the sine/cosine Fourier series and the coefficients \(f_n\) in the complex exponential Fourier series: \[\begin{align} f(x) &= \sum_{n=1}^\infty \alpha_n \sin\left(\frac{2\pi n x}{a}\right) + \sum_{m=0}^\infty \beta_m \cos\left(\frac{2 \pi m x}{a}\right) \\ &= \sum_{n=-\infty}^\infty f_n \exp\left(\frac{2\pi i n x}{a}\right). \end{align}\]

Exercise \(\PageIndex{2}\)

Consider the triangular wave \[f(x) = \left\{\begin{array}{rr}- x, &-a/2 \le x < 0, \\ x, & 0 \le x < a/2\end{array}\right.\]

1. Derive the Fourier series expansion.

2. Plot the Fourier series numerically, and show that it converges to the triangular wave as the number of terms increases.

Exercise \(\PageIndex{3}\)

A periodic function \(f(x)\) (with period \(a\)) is written as a complex Fourier series with coefficients \(\{f_0, f_{\pm1}, f_{\pm2}, \dots\}\). Determine the relationship(s) between the Fourier coefficients under each of the following scenarios:

1. \(f(x)\) is real for all \(x\).

2. \(f(x) = f(-x)\) for all \(x\)

3. \(f(x) = f(-x)^*\) for all \(x\).

Answer

The Fourier coefficients are given by \[f_n = \frac{1}{a} \int_{-a/2}^{\,a/2} dx\; e^{-i k_n x}\, f(x), \quad \mathrm{where}\;\, k_n = \frac{2\pi n}{a}.\] First, consider the case where \(f(x)\) is real. Take the complex conjugate of both sides: \[\begin{align} f_n^* &= \frac{1}{a} \int_{-a/2}^{\,a/2} dx\; \left(e^{-i k_n x}\, f(x)\right)^* \\ &= \frac{1}{a} \int_{-a/2}^{\,a/2} dx\; e^{i k_n x}\, f(x)^* \\ &= \frac{1}{a} \int_{-a/2}^{\,a/2} dx\; e^{i k_n x}\, f(x) \\ &= f_{-n}.\end{align}\] Hence, \[f_{n} = f_{-n}^*.\] For the second case, \(f(x) = f(-x),\) perform a change of variables \(x = -u\) in the Fourier integral: \[\begin{align} f_n &= \frac{1}{a} \int_{-a/2}^{\,a/2} du\; e^{i k_n u}\, f(u) \\ &= f_{-n}.\end{align}\] For \(f(x) = f(-x)^*\), the same change of variables gives \[f_n = f_n^*.\]

Exercise \(\PageIndex{4}\)

Prove the properties of the Fourier transform listed in Section 10.4.

Exercise \(\PageIndex{5}\)

Find the Fourier transform of \(f(x) = \sin(\kappa x)/x.\)

Exercise \(\PageIndex{6}\)

Prove that if \(f(x)\) is a real function, then its Fourier transform satisfies \(F(k) = F(-k)^*\).

Exercise \(\PageIndex{7}\)

Prove that \[\delta(ax) = \frac{1}{a}\,\delta(x),\] where \(a\) is any nonzero real number.

Answer

From the definition of the delta function as the narrow-peak limit of a Gaussian wavepacket: \[\delta(ax) = \lim_{\gamma \rightarrow 0} \, \int_{-\infty}^\infty \frac{dk}{2\pi} \, e^{ikax} \, e^{-\gamma k^2}.\] Perform a change of variables \(k = q/a\) and \(\gamma = \gamma' \, a^2\): \[\begin{align} \delta(ax) &= \lim_{\gamma' \rightarrow 0} \, \int_{-\infty}^\infty \frac{1}{a}\frac{dq}{2\pi} \, e^{iqx} \, e^{-\gamma' q^2} \\ &= \frac{1}{a} \delta(x).\end{align}\]

Exercise \(\PageIndex{8}\)

Calculate \[\int_{-\infty}^\infty dx \int_{-\infty}^\infty dy \; x^2\, \delta\left(\sqrt{x^2+y^2}-a\right),\] where \(a\) is a real number.

Answer

Perform a change of variables from Cartesian coordinates \((x,y)\) to polar coordinates \((r,\phi)\): \[\begin{align} \int_{-\infty}^\infty dx \int_{-\infty}^\infty dy \; x^2\, \delta\left(\sqrt{x^2+y^2}-a\right) &= \int_0^{\infty} dr \int_{0}^{2\pi} rd\phi\, \cdot\, r^2\cos^2\phi\; \delta(r-a) \\ &= \left(\int_0^{\infty} dr \, r^3\, \delta(r-a)\right) \left(\int_{0}^{2\pi}\!d\phi \, \cos^2\phi\right) \\ &= \begin{cases}\pi a^3, & a \ge 0 \\ 0, & a < 0.\end{cases} \end{align}\]