Homework 3
Instructions: Scan your solution and upload it to Canvas. Your solution does not have to be
typed on a computer but it has to be clear, legible and in a single pdf file.
You should include the Mathematica code that you use as well (Export it as a PDF and merge
it with your solution).
This portion is to be done by hand. You should include all the details.

Problem 1.

Find a non-zero quadratic polynomial that is orthogonal to $x, x^{2}$ and $x^{3}$ on the interval $[-1,1]$.

Proof .

start by assuming that the polynomial that we are seeking is $p(x)=a x^{2}+b x+c$ and find $a, b, c$.

Problem 2.

Compute the Fourier series of the following functions on $(-\pi, \pi)$, you should simplify the coefficients.
(a) $x^{2}$.
(c) $\cos (2 x)$
(b) $e^{x}$.
(d) $|\sin (x)|$

Proof .

In (c) and (d), you will see that the coefficients $a_{n}$ and/or $b_{n}$ are undefined at certain values of $n$. You will have compute these coefficients separately， and this is note hard.

Problem 3.

Let $f$ be defined on $(-\pi, \pi)$ by

(a) Compute and simplify $S(x)$ (the Fourier series of $f$ ).
(b) Simplify the expression of $S(\pi)$. (It will involve a sum).
(c) Using Dirichlet’s theorem, find $S(\pi)$. (Hint: think about the periodic extension of $f$, what is $f\left(\pi^{-}\right)$ and $\left.f\left(\pi^{+}\right)\right)$.
(d) Use (b) and (c) to show that
$$\frac{\pi^{2}}{6}=\sum_{n=1}^{\infty} \frac{1}{n^{2}}=1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\ldots$$

Proof .

This is known as Basel problem, it was solved by Euler in 1734. This formula catapulted
Euler to fame because the mathematical community has never seen something like it before.
There are numerous Youtube videos and articles about how Euler approached this problem and
about other ways to solve it, for example see 3Blue1Brown.

Instructor: Martin Klaus ([email protected])

Office: McBryde 472, phone 231-6533

Office Hours: MW: 10 am – 11:30 am, or by appointment

Text: Elementary Differential Equations with Boundary Value Problems, second edition, by W. Kohler and L. Johnson

Prerequisites: Math 2214

Course Content: The topics covered in this course include the Laplace transform, partial differential equations and separation of variables, Fourier series, and if time permits, the Fourier transform and Sturm-Liouville theory.

Evaluation Policy: Your grade will be based on homework assignments, quizzes, two in-class exams, and the final exam. The homework is worth 10%. The quizzes together are 30%, the in-class tests each count for 20%, and the final exam is worth 20% of your grade.

A score of 90% will guarantee an A-, 80% a B-, 70% a C-, and 60% a D-.

Quiz dates: TBA

Test dates: Friday, June 14

Final Exam: Monday, May 11, 7:45 am – 9:45 am

Homework: Homework will be assigned on a regular basis and must be turned in on some due date at the beginning of class. The homework will be graded. You are allowed to discuss the problems with me and/or other students while you are working on them. However, the work you turn in must represent your understanding of the material and it must be written up independently. Late homework will only be accepted in special circumstances upon consultation with me.

Absences: Make-up exams will generally not be given. If a student has a valid excuse (e.g. documented illness) the student should contact me before the exam.

Virginia Tech Honor System: I will assume that you have read the Va Tech Honor policy (https://www.honorsystem.vt.edu) and I expect you to abide by it. The Undergraduate Honor Code pledge that each member of the university community agrees to abide by states: “As a Hokie, I will conduct myself with honor and integrity at all times. I will not lie, cheat, or steal, nor will I accept the actions of those who do.” A student who has doubts about how the Honor Code applies to any assignment is responsible for obtaining specific guidance from the course instructor before submitting the assignment for evaluation. Ignorance of the rules does not exclude any member of the University community from the requirements and expectations of the Honor Code.