10.9: Exercises

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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 $$x$$.

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.

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.

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}