代写Math 6B Worksheet 9 Winter 2025代写数据结构语言

Math 6B

Worksheet 9

Winter 2025

Due Saturday, Mar 8, at 11:59pm.

1. Let f(x) = x 2 on [−1, 1] be given.

(a) Find a formula for the Fourier series of this function.

(b) Using the Fourier series you obtained in (a), show that

2. Show that the trigonometric identity

could be interpreted as a Fourier series. Use this identity to obtain the Fourier series of cos4 x without finding the Fourier series coefficients directly.

3. Use the Fourier series of the function f(x) = |sin x| on [−π, π] to evaluate the following series

4. Show that the differentiation turns an odd function to an even function and an even function to an odd function.

5. Consider the Fourier series of the function f(x) = x on [0, l]. Assume the series could be integrated term by term (this has to be justified since the series is infinite).

(a) What is the Fourier cosine series of ? What is the constant of integration, which is the first term in the series?

(b) Let x = 0. Evaluate the series

6. Let f(x) be a periodic function with T = π. Let

for all x. Find the coefficients an.

7. Show that by the change of variable we are able to derive the Fourier series on [−l, l] from the Fourier series on [−π, π].

8. Use the trigonometric polynomial, obtained from Fourier series, with N = 1, 2, 3, 4, 5 to find the error estimate of the function f(x) = |x| on [−π, π].

9. Conversion to Sturm-Liouville form. Consider

Show that with we are able to convert the equation above to the Sturm-Liouville form.