Reference no: EM131103835
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 bn = 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 = limx→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)DN(x - t)dt = 1/2L-L∫Lf(x - t)DN(t)dt, where DN(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∫LDN(x)dx
10. Show that
limt→0(f(x - t) - f(x)/2 sin πt/2L) = L/π f'(x)
and that therefore, limt→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
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