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MATH 3C03: Home Assignment # 3

Problem 1: Consider Chebyshev’s (type II) differential equation

Chebyshev’s polynomials of the second kind are defined by the recurrence rela-tion

starting with and Obtain the first five polynomials from the recurrence relation. Specify the orthogonality condition between the polynomials for n  m. Compute their norm defined by the inner product:

Problem 2: Expand for in terms of the eigenfunctions of the boundary-value problem

Compute coefficients of the series of eigenfunctions explicitly.

Problem 3: Compute coefficients of the Fourier series of the function f given by

Since explain in what sense the Fourier series converges to f. Justify why the Fourier series at x = 0 converges to f(0) = 0 and confirm that

Problem 4: Integrate the Fourier series in Problem 3 term by term to compute the Fourier series for

Be careful when presenting the Fourier series for g(x) since the Fourier series should only contain sine and cosine functions. Explain why the Fourier series for g(x) cannot be differentiated term by term in order to recover back f(x).

Problem 5: Suggest a polynomial on [0, π] whose Fourier coefficients decay as rapidly as  and confirm the decay by computing the Fourier coefficients explicitly.

Hint: if Fourier coefficients decay as rapidly as , then the Fourier series can be differentiated term by term twice.