Chapter 21, Spherical functions Video Solutions, Differential equations of mathematical physics

Problem 1

Show that the integral

$$
\int_{-\pi}^\pi f\left(x_3+i x_1 \cos \zeta+i x_2 \sin \zeta, \zeta\right) d \zeta
$$

where $f(\xi, \zeta)$ is an arbitrary function that can be twice differentiated under the integral sign with respect to the parameters $x_1, x_2$, and $x_3$, is a solution to Laplace's equation.

Will Erickson

Numerade Educator

Suppose that $\gamma$ is the angle between two radii drawn from the center of a spherical surface $\Sigma$ to the points $\left(\theta_{\mathrm{O}}, \varphi_{\mathrm{O}}\right)$ and $(\theta, \varphi)$ on $\Sigma$. Assuming that $\gamma=\gamma(\theta, \varphi)$, show that

$$
\begin{aligned}
P_n(\cos \gamma)=P_n(\cos \theta) P_n\left(\cos \theta_{\mathrm{O}}\right) & \\
& \quad+\sum_{k=0}^n 2 \frac{(n-k)!}{(n+k)!} P_{n k}(\cos \theta) P_{n k}\left(\cos \theta_{\mathrm{O}}\right) \cos k\left(\varphi-\varphi_{\mathrm{O}}\right),
\end{aligned}
$$

which is known as the composition theorem for Legendre polynomials.
Method: Expand $P_n(\cos \gamma)$ in a series of the form (30) and use formulae (23) for calculating the coefficients in the series.

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00:51

Problem 1

Find the solution to the interior Dirichlet problem in the form (33), starting with Poisson's integral (51) of Chapter VIII.
Method: Use the expansion

$$
\frac{1-h^2}{\left(1-2 h \cos \theta+h^2\right)^{\frac{8}{2}}}=\sum_{k=0}^{\infty}(2 k+1) h^k P_k(\cos \theta) \quad(h<1)
$$

Hast Aggarwal

Numerade Educator

Problem 1

Starting with expression (42) for the Green's function, derive Poisson's integral formula (51) of Chapter XVIII.

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00:59

Problem 1

Show that the Green's function of the Neumann problem stated for an infinite region lying outside some sphere is expressed by
$$
G(\xi, x)=\frac{1}{r}+\frac{a}{|x| r_1}+\frac{1}{a} \ln \frac{(1-\cos \gamma)|x||\xi|}{a^2+r_1|x|-|x||\xi| \cos \gamma} .
$$

Method: Expand the function $\varphi$ appearing in the relationship (40) in a series of Legendre polynomials:

$$
\varphi=\sum_{k=0}^{\infty} a_k P_k(\cos \gamma)\left(\frac{a^2}{|x||\xi|}\right)^{k+1},
$$

and use the boundary conditions (44). In summing the series, use the formula obtained by integrating the equation

$$
\frac{1}{\sqrt{1+\rho^2-2 \rho \cos \gamma}}=\sum_{k=0}^{\infty} P_k(\cos \gamma) \rho^k \quad(|\rho|<1)
$$

Raj Bala

Numerade Educator

01:00

Problem 2

Show that all second-order tesseral spherical functions can be obtained by twice differentiating $1 / R$ in the directions of the coordinate axes.

Raj Bala

Numerade Educator

Problem 2

Find the solution to the interior Dirichlet problem with the boundary condition

$$
\left.u(R, \theta, \varphi)\right|_{R=R_0}=\sin 3 \theta \cos \varphi
$$

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Problem 2

Use the solution to problem 1 to show that the solution to the exterior Neumann problem for a sphere $V$ is given by the following Bjerknesformula:

$$
u(x)=\frac{1}{4 \pi} \iint_{\mathscr{F} V}\left[\frac{2}{r}-\frac{1}{a} \ln \frac{a+r-|x| \cos \gamma}{(1-\cos \gamma)|\xi|}\right] \psi \mathrm{d} S
$$

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01:00

Problem 3

Show that all tesseral spherical functions can be obtained from the function $1 / R$ by differentiating it $n-m$ times in the direction $x_3$ and $m$ times in the directions lying in the 1-2 plane at an angle $\pi / m$ to each other.

Raj Bala

Numerade Educator

Problem 3

Demonstrate the possibility of solving the mixed boundary problem for a spherical surface by means of spherical functions.

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Problem 4

The curves of a spherical surface along which the value of a spherical function is equal to zero are called the nodal curves of that spherical function.
(a) Show that the nodal lines of the Legendre polynomial $P_n(\cos \theta)$ represent the parallels dividing the spherical surface into $n+1$ zones characterized by the fact that, in each of these zones, $P_n(\cos \theta)$ retains its sign, changing sign only upon crossing a nodal line.
(b) Show that the nodal lines of the tesseral spherical functions

$$
P_{n m}(\cos \theta) \cos m \varphi \quad \text { and } \quad P_{n m}(\cos \theta) \sin m \varphi
$$

constitute $n$ parallels and $m$ equally spaced meridians which partition the spherical surface into cells (tesserae) characterized by the fact that these functions retain their sign throughout each of the cells but change it on crossing the boundary of a cell (that is, a nodal line) (see fig. 54).

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Problem 4

Solve Dirichlet's problem for the region between two concentric spherical surfaces with radii $R_1$ and $R_2$ with the condition that the desired solution becomes the given function $f(\theta, \varphi)$ on the first spherical surface and the function $F(\theta, \varphi)$ on the second.

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26:49

Problem 5

Solve the preceding problem under the assumption that the functions $f$ and $F$ depend only on the angle $\theta$.

Khoobchandra Agrawal

Numerade Educator