Chapter 13: Problems

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13.1 Problems

1. (Boas Chapter 10, Section 11, Problems 5, 7, 8.) Consider thecoordinates $(u,v,z)$ defined by $x=u(1-v)$, $y = u\sqrt{2v-v^2}$,$z=z$.

   a) Find the basis vectors $\boldsymbol{\hat{u}}$ and $\boldsymbol{\hat{v}}$, the scale factors $h_u$ and $h_v$, and the infinitesimal displacement vector $d\boldsymbol{r}$.

   b) Write expressions for $\boldsymbol{\nabla} f$, >$\boldsymbol{\nabla} \cdot \boldsymbol{A}$, and $\nabla^2 f$.

   c) Calculate $\boldsymbol{\nabla} \cdot \boldsymbol{\hat{u}}$, $\boldsymbol{\nabla} \times \boldsymbol{\hat{v}}$, and $\nabla^2 \ln u$.

2. (Boas Chapter 11, Section 13, Problem 23.) Express $\Gamma(55.5)$ in terms of powers of 2, a factor of $\sqrt{\pi}$, and the double factorial of a number.

3. Show that the Gamma function can be expressed as $\Gamma(x) = 2 \int_0^{\infty} t^{2x-1} e^{-t^2}\, dt$.

4. (Boas Chapter 12, Section 23, Problem 1.) Use the generating function to prove that the Legendre polynomials are normalized by $\int_{-1}^1 [P_\ell(x)]^2\, dx = 2/(2\ell+1)$. You may proceed as follows. First, square the relation $\Phi(x,h) = \sum_\ell P_\ell(x) h^\ell$ and integrate both sides from $x=-1$ to $x=1$. Second, invoke the series expansion $\ln(1+h) = h - h^2/2 + h^3/3 - h^4/4 + h^5/5 + \cdots$ to match the powers of $h$ on each side of the equation.

5. Prove the identity

   \[\frac{1 - h^2}{(1 - 2x h + h^2)^{3/2}} = \sum_{\ell=0}^\infty (2\ell + 1) P_\ell(x)\, h^\ell.\]

6. Express $\delta(x)$ as an expansion in Legendre polynomials in the interval $-1 \leq x \leq 1$.

7. Express $x^6$ as an expansion in Legendre polynomials.

8. Prove that $\int_{-1}^1 (1-x^2)^{-1} P_\ell^m(x) P_\ell^{m'}(x)\, dx = 0$ when $m \neq m'$.

9. Express $\delta(\theta-\theta') \delta(\phi-\phi')/\sin\theta$ as an expansion in spherical harmonics.

10. (Boas Chapter 12, Section 23, Problem 10.) Evaluate $\int_0^\infty x^{-p} J_{p+1}(x)\, dx$.

11. (Boas Chapter 12, Section 23, Problem 14.) Evaluate $\int x^3 J_0(x)\, dx$.

12. Prove that $J_0(x) = (2/\pi) \int_0^1 (1-t^2)^{-1/2} \cos(xt)\, dt$.

13. Prove that if $f(x)$ goes to zero at $x = x_j$ ($j = 1, 2, 3,

    \cdots$), then

    \[\delta\bigl( f(x) \bigr) = \sum_j | f'(x_j) |^{-1} \delta(x-x_j),\]

    where a prime indicates differentiation with respect to $x$.

14. (Boas Chapter 7, Section 13, Problem 6.) A periodic function $f(x)$ is defined to be equal to $e^{ikx}$ in the interval $-\pi \leq x \leq \pi$ and to repeat such that $f(x+2\pi) = f(x)$; $k$ is not equal to an integer. Express this function as a Fourier series.

15. A periodic function $f(x)$ is defined to be equal to $0$ in the interval $-\pi \leq x \leq 0$, to $\sin x$ in the interval $0 < x \leq \pi$, and to repeat such that $f(x+2\pi) = f(x)$. Express this function as a Fourier series.

16. (Boas Chapter 7, Section 13, Problem 15.) A function $f(t)$ is defined to be equal to $1$ in the interval $-2 \leq t \leq 0$, to $-1$ in the interval $0 < t \leq 2$, and to $0$ outside of these intervals. Calculate its Fourier transform $g(\omega)$.

17. A function $f(t)$ is defined to be equal to $t^{-1/2}$ when $t > 0$ and to $0$ when $t \leq 0$. Calculate its Fourier transform $g(\omega)$.

18. (Boas Chapter 13, Section 10, Problem 1.) Solve the two dimensional Laplace equation in the domain $0 \leq x \leq 1$, $0 \leq y \leq 2$, with the boundary conditions $V(x=0,y) = 0$, $V(x=1,y) = 0$, $V(x,y=0) = 1 - x$, $V(x,y=2) = 0$.

19. (Boas Chapter 13, Section 10, Problem 14.) An infinitely long cylinder of radius $1$ has been cut into quarter cylinders that are insulated from each other. Alternate quarter cylinders are held a potentials $+100$ and $-100$. Find the potential inside the cylinder.

20. (Boas Chapter 13, Section 10, Problem 21.) The surface of a sphere of radius $1$ is maintained at the potential $V = \sin^2\theta + \cos^3\theta$. What is the potential inside the sphere?

21. A sphere of radius $a$ is surrounded by another sphere of radius $b$; the spheres are concentric. On the surface of the smaller sphere (radius $a$) the potential is $V = V_0 \cos\theta$. On the surface of the larger sphere (radius $b$) the potential is $V = V_1 \sin\theta \cos\phi$. What is the potential between the spheres?

22. (Boas Chapter 13, Section 10, Problem 9.) A string with fixed ends at $x=0$ and $x=L$ is given an initial displacement $f(x) = x(L-x)$ and an initial velocity $g(x) = 0$. Find the lateral displacement $\psi(x,t)$ at any other time $t$.

23. A string with fixed ends at $x=0$ and $x=\pi$ is given an initial displacement $f(x) = f_0 \sin x$ and an initial velocity $g(x) = g_0 \sin x$. Find the lateral displacement $\psi(x,t)$.

24. An infinitely long string is given an initial displacement $f(x) = \sin(3 x)$ and an initial velocity $g(x) = v \sin(3x)$. Find the lateral displacement $\psi(t,x)$.

25. A square membrane of side $a = b = 1$ is given an initial displacement $f(x,y) = f_0 \sin^2(\pi x) \sin^2(\pi y)$ and an initial velocity $g(x,y) = 0$. Find the lateral displacement $\psi(t,x,y)$.

26. A circular membrane of radius $1$ is given an initial displacement $f(s,\phi) = s(1-s^2) \cos\phi$ and an initial velocity $g(s,\phi) = 0$. What is the lateral displacement $\psi(t,r,\phi)$ at any other time $t$?

27. (Boas Chapter 8, Section 13, Problem 48.) Solve the inhomogeneous differential equation $d^2X/dt^2 + X = t \sin t$ for the function $X(t)$. The initial conditions are that $X = dX/dt = 0$ at $t=0$.

13.2 Answers to Problems

1. \begin{align} & \boldsymbol{\hat{u}} = (1-v)\, \boldsymbol{\hat{x}} + \sqrt{2v-v^2}\, \boldsymbol{\hat{y}}, \qquad \boldsymbol{\hat{v}} = - \sqrt{2v-v^2}\, \boldsymbol{\hat{x}} + (1-v)\, \boldsymbol{\hat{y}} \nonumber \\ & h_u = 1, \qquad h_v = \frac{u}{\sqrt{2v-v^2}}, \qquad d\boldsymbol{r} = du\, \boldsymbol{\hat{u}} + \frac{u\, dv}{\sqrt{2v-v^2}}\, \boldsymbol{\hat{v}} + dz\, \boldsymbol{\hat{z}} \nonumber \\ & \boldsymbol{\nabla} f = \frac{\partial f}{\partial u}\, \boldsymbol{\hat{u}} + \frac{\sqrt{2v-v^2}}{u}\, \frac{\partial f}{\partial u}\, \boldsymbol{\hat{v}} + \frac{\partial f}{\partial z}\, \boldsymbol{\hat{z}} \nonumber \\ & \boldsymbol{\nabla} \cdot \boldsymbol{A} = \frac{1}{u} \frac{\partial}{\partial u} \Bigl( u A_u \Bigr) + \frac{\sqrt{2v-v^2}}{u} \frac{\partial A_v}{\partial v} + \frac{\partial A_z}{\partial z} \nonumber \\ & \nabla^2 f = \frac{1}{u} \frac{\partial}{\partial u} \biggl( u \frac{\partial f}{\partial u} \biggr) + \frac{\sqrt{2v-v^2}}{u^2} \frac{\partial}{\partial v} \biggl( \sqrt{2v-v^2} \frac{\partial f}{\partial v} \biggr) + \frac{\partial^2 f}{\partial z^2} \nonumber \\ & \boldsymbol{\nabla} \cdot \boldsymbol{\hat{u}} = \frac{1}{u}, \qquad \boldsymbol{\nabla} \times \boldsymbol{\hat{v}} = \frac{1}{u}\, \boldsymbol{\hat{z}}, \qquad \nabla^2 \ln u = 0 \nonumber \end{align}

2. \[\Gamma(55.5) = \frac{109!!}{2^{55}} \sqrt{\pi}\]

3. Change the variable of integration.

4. Follow the instructions.

5. Involve the generating function and its derivative with respect to $h$.

6. \[\delta(x) = \frac{1}{2} \sum_{\ell=0,2,4,\cdots}^\infty (2\ell+1) P_\ell(0)\, P_\ell(x) \]

7. \[x^6 = \frac{1}{7} P_0 + \frac{10}{21} P_2 + \frac{24}{77} P_4 + \frac{16}{231} P_6 \]

8. Start from the differential equation satisfied by the associated Legendre functions, and proceed as for the proof of orthogonality for functions with different values of $\ell$.

9. \[ \frac{\delta(\theta-\theta') \delta(\phi-\phi')}{\sin\theta} = \sum_{\ell m} \bigl[ Y_\ell^m(\theta',\phi') \bigr]^* Y_\ell^m(\theta,\phi)\]

10. \[\frac{1}{2^p p!}\]

11. \[x^3 J_1(x) - 2x^2 J_2(x)\]

12. Start from the integral representation, and consider a change of variable of integration.

13. The equality is established by integrating both sides against a test function $g(x)$. Break the domain of integration into a number of segments, each one beginning at a local minimum (or local maximum) of $f$ and ending at the next local maximum (or local minimum); each segment necessarily contains a single zero $x_j$. Then change the variable of integration.

14. \[f(x) = \frac{1}{\pi} \sin(k\pi) \sum_{n=-\infty}^\infty \frac{(-1)^n}{k-n} e^{inx}\]

15. \[f(x) = \frac{1}{\pi} + \frac{1}{2} \sin x - \frac{2}{\pi} \sum_{n=2,4,6,\cdots}^\infty \frac{\cos(nx)}{n^2-1}\]

16. \[g(\omega) = \frac{1-\cos(2\omega)}{i\pi \omega}\]

17. \[g(\omega) = \left\{ \begin{array}{ll} \frac{i+i}{2\sqrt{2\pi}} \omega^{-1/2} & \quad \omega > 0 \\ \frac{i-i}{2\sqrt{2\pi}} |\omega|^{-1/2} & \quad \omega < 0 \end{array} \right.\]

18. \[V(x,y) = \frac{2}{\pi} \sum_{n=1}^\infty \frac{1}{n \sinh(2n\pi)} \sin(n\pi x) \sinh\bigl[ n\pi(2-y) \bigr]\]

19. \[V(s,\phi) = \frac{800}{\pi} \sum_{m=2,6,10,\cdots}^\infty \frac{1}{m} s^m \sin(m\phi)\]

20. \[V(r,\theta) = \frac{2}{3} P_0(\cos\theta) + \frac{3}{5} r P_1(\cos\theta) - \frac{2}{3} r^2 P_2(\cos\theta) + \frac{2}{5} r^3 P_3(\cos\theta)\]

21. \[ V(r,\theta,\phi) = \frac{V_0}{1-a^3/b^3} \frac{a^2}{r^2} \biggl( 1 - \frac{r^3}{b^3} \biggr) \cos\theta + \frac{V_1}{1-a^3/b^3} \frac{r}{b} \biggl( 1 - \frac{a^3}{r^3} \biggr) \sin\theta \cos\phi\]

22. \[\psi(t,x) = \frac{8L^2}{\pi^3} \sum_{n=1,3,5,\cdots}^\infty \frac{1}{n^3} \sin \Bigl( \frac{n\pi x}{L} \Bigr) \cos \Bigl( \frac{n\pi vt}{L} \Bigr)\]

23. \[\psi(t,x) = f_0 \sin x \cos vt + \frac{g_0}{v} \sin x \sin vt\]

24. \[\psi(t,x) = \frac{1}{2} \sin(3x - 3vt) + \frac{1}{2} \sin(3x + 3vt) + \frac{1}{6} \cos(3x - 3vt) - \frac{1}{6} \cos(3x + 3vt) \]

25. \[\psi(t,x,y) = \frac{64 f_0}{\pi^2} \sum_{n=1,3,5,\cdots}^\infty \sum_{m=1,3,5,\cdots}^\infty \frac{1}{nm(n^2-4)(m^2-4)} \sin(n\pi x) \sin(m\pi y) \cos\Bigl( \pi \sqrt{n^2+m^2}\, v t \Bigr) \]

26. \[ \psi(t,s,\phi) = 4\cos\phi \sum_{p=1}^\infty \frac{J_3(\alpha_{1p})}{\alpha_{1p}^2 \bigl[ J_2(\alpha_{1p})\bigr]^2} J_1(\alpha_{1p} s) \cos( \alpha_{1p} vt )\]

27. \[X(t) = \frac{1}{4} t \sin t - \frac{1}{4} t^2 \cos t\]