MATH 4340, Section 1, Homework Assignment 6 Phase Angle Representations, Complex Fourier Series, and Eigenvalues Due: 9:05 am, Monday, March 7, 2016 Point value: 20 The purpose of this homework assignment is to help you 1. work with phase angle representations of Fourier series; 2. compute and plot complex Fourier series representations; 3. recall eigenvalues and eigenvectors from linear algebra (or ordinary dierential equations). Each individual is expected to complete his or her own assignment. Show all work and justify all conclusions. No late homework will be accepted. 1. Let f (x) = x2 , x [0, 2), f (x) = f (x + 2). (a) (3 pts.) Use the Fourier series expansion of f (x) to generate the complex Fourier series representation for f (x). That is, explicitly use the coecients an and bn in the Fourier series representation to generate cn . (b) (4 pts.) Directly compute the values of cn needed for the complex Fourier series expansion of f (x). Compare your answer with part (a). Be sure to incorporate Euler's identity when necessary. For instance, note the simplication ein = cos (n) + i sin (n) = (1)n . (c) (5 pts.) Write the sine and cosine phase angle forms of the Fourier series expansion for f (x). You do not have to explicitly nd n , but you should dene it in terms of its arctangent formulation and provide the appropriate quadrant. 2. (4 pts.) Find the eigenvalues and eigenvectors of the matrix 5 8 16 1 8 . A= 4 4 4 11 Use the fact that = 1 is an eigenvalue to help you nd the remaining two eigenvalues. Determine whether or not the eigenvectors are linearly independent and/or orthogonal. 3. (4 pts.) Find the eigenvalues and eigenvectors 3 A= 2 4 of the matrix 2 4 0 2 . 2 3 (a) Find the eigenvalues and associated eigenvectors, using the fact that = 1 is an eigenvalue. There are two eigenvectors associated with = 1 and one with the third eigenvalue; choose the eigenvectors so they form an orthogonal set. (b) Use the orthogonality to write v = 3i4j+k as a linear combination of the eigenvectors. 1