Problem 1.

1.

(a) The integral representation for the function $J_{2}(r)$ is
$$J_{2}(r)=\frac{1}{2 \pi} \int_{0}^{2 \pi} e^{i r \sin (\theta)} e^{2 i \theta} d \theta$$
Using this form, and any identities about Dirac delta functions you find useful, find the Fourier transform of this function, $\tilde{J}{2}(k)$.

(b) The functions $f(x)$ and $g(x)$ are defined by

and $g(x)=f(-x)$. Find the function $h(x)$, defined by $h(x)=\int_{-\infty}^{\infty} f(x-y) g(y) d y .$

Proof .

Problem 2.

1. (a) Find the Green’s function for the equation
$$y^{\prime \prime}-\frac{2}{\sin ^{2} x} y=f(x) \quad 0<x<\pi / 2$$
with the boundary conditions $y(0)=y(\pi / 2)=0 .$ The solutions to the homogeneous equation are $\cot x$ and $1-x \cot x$.
(b) Use the Green’s function to find the particular solution to this differential equation that satisfies the given boundary conditions, if $f(x)=\sin ^{2}(x)$. You do not need to evaluate any integrals that you end up with.

Proof .

Problem 3.

3.

A hollow sphere of radius 5 is cut out of an infinite metal block. The temperature on the surface of the sphere is maintained at $u(r=5, \theta)=$ $100 \sin ^{2} \theta .$ Find the temperature profile $u(r, \theta)$ in the metal block when the system reaches steady state.
Legendre polynomials:

$P_{0}(x)=1$

$P_{1}(x)=x$

$P_{2}(x)=\left(3 x^{2}-1\right) / 2$

$P_{3}(x)=\left(5 x^{3}-3 x\right) / 2$

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# 复分析代考

## Math 224a: Mathematical Methods for the Physical Sciences. Fall 2021.

InstructorNikhil Srivastava, email: firstname at math.obvious.edu

Lectures: TTh 2:10-3:30pm, Hildebrand B56.

Office Hours: T: 3:40-4:40pm, Th 5-6pm (1035 Evans)

Textbooks. Much of the material will be drawn from the following two books. Both are available to Cal students for free on Oskicat or Springerlink.
Robert Richtmyer, Principles of Advanced Mathematical Physics, Volume I
Reed and Simon, Functional Analysis, Vol I
Jonathan Dimock, Quantum mechanics and quantum field theory [electronic resource] : a mathematical primer
Folland, Intro to PDE, 2e
Trefethen, Approximation Theory and Approximation Practice
I will also draw on several other resources and frequently post lecture notes on this webpage.

Announcements

• (9/4) Office hours this week: Th 5-6 and F 4-5pm.
• (9/24) Office thours this week: W 4-5 instead of Th.
• (10/3) Lecture today is cancelled due to sickness.
• (11/12) Office hours this week: F 4-5pm instead of T 3:40-4:40.

Syllabus The course will survey methods for solving the fundamental problems of mathematical physics. The overall purpose of the course will be to develop a functional analytic framework for understanding and approximating solutions of differential equations, with an emphasis on physical examples. The content can broadly be divided into three parts:

1. Functional Analysis. (3 weeks) Lp spaces, Hilbert spaces, distributions, Schwartz functions, Fourier transform, potential theory.
2. Spectral Theory. (7 weeks) Linear operators, adjoint, spectrum and resolvent, spectral theorem for bounded s.a. operators, Fredholm alternative, Green’s functions, Sturm-Liouville theory.
3. Orthogonal Polynomials. (4 weeks) Classical orthogonal polynomials; approximation and interpolation theory.

### Class Schedule

Homework. Will be due every two weeks, on Thursday in class. HW assignments will be updated (i.e., problems may be added) until upto a week before they are due. Please write clearly or type your solutions using Latex. Collaboration is allowed but you must list your collaborators in your writeup.

1. HW1, due 9/12.
2. HW2, due 9/26.
3. HW3, due 10/10.
4. HW4, due 10/31.
5. HW5, due 11/21.
6. HW6, due 12/17.

Grading. 100% homework. The bottom assignment will be dropped.