Math 054 partial differential equations - hw assignment 4

Question

Math 054 Partial Differential Equations - HW Assignment 4

1. Find the Fourier Series for f(x) = x if -π < x < π.

2. Find the Fourier Series for f(x) = x² if -1 < x < 1.

3. Find the Fourier Series for

 

4. Find the Fourier Series for f(x) = |x| if -1 < x < 1.

5. Suppose that f is 2L- periodic and has the Fourier series representation. Prove that f is even if and only if b_n = 0 for all n.

6. Fourier claimed that all functions could be represented as a Fourier series. Is this true? Explain.

7. The Dirichlet Kernel for any real number x ≠ 2kL is given by,

1 + 2 cos (π/L)x + 2 cos 2(π/L)x + ... + 2 cos N(π/L)x = (sin ((N + ½)nπ/L)/sin πx/2L)

Show that for x = 2kL,

1 + 2 cos (π/L)x + 2 cos 2(π/L)x + ... + 2 cos N(π/L)x = lim_x→2kL(sin ((N+1/2)nπ/L)/sin πx/2L)

8. Using the change of variables T = x - t, show

1/2L_-L∫^Lf(t)D_N(x - t)dt = 1/2L_-L∫^Lf(x - t)D_N(t)dt, where D_N(t) is the Dirichlet Kernel.

9. Prove the following Dirichlet properties,

(a) The Dirichlet Kernel is even for all x.

(b) For all N = 1, 2, ..., we have

1/2L_-L∫^LD_N(x)dx

10. Show that

lim_t→0(f(x - t) - f(x)/2 sin πt/2L) = L/π f'(x)

and that therefore, lim_t→0g(t) = g(0) where

11. Find the Fourier Series for f(x) = sin(x) if -π < x < π.

12. Find the Fourier Series for