 10.2.1: In each of 1 through 8, determine whether the given function is per...
 10.2.2: In each of 1 through 8, determine whether the given function is per...
 10.2.3: In each of 1 through 8, determine whether the given function is per...
 10.2.4: In each of 1 through 8, determine whether the given function is per...
 10.2.5: In each of 1 through 8, determine whether the given function is per...
 10.2.6: In each of 1 through 8, determine whether the given function is per...
 10.2.7: In each of 1 through 8, determine whether the given function is per...
 10.2.8: In each of 1 through 8, determine whether the given function is per...
 10.2.9: If f ( x) = x for L < x < L, and if f ( x + 2L) = f ( x), find a fo...
 10.2.10: If f ( x) = x + 1, 1 < x < 0, x, 0< x < 1, and if f ( x + 2) = f ( ...
 10.2.11: If f ( x) = L x for 0 < x < 2L, and if f ( x + 2L) = f ( x), find a...
 10.2.12: Verify equations (6) and (7) in this section by direct integration.
 10.2.13: In each of 13 through 18: a. Sketch the graph of the given function...
 10.2.14: In each of 13 through 18: a. Sketch the graph of the given function...
 10.2.15: In each of 13 through 18: a. Sketch the graph of the given function...
 10.2.16: In each of 13 through 18: a. Sketch the graph of the given function...
 10.2.17: In each of 13 through 18: a. Sketch the graph of the given function...
 10.2.18: In each of 13 through 18: a. Sketch the graph of the given function...
 10.2.19: In each of 19 through 24: a. Sketch the graph of the given function...
 10.2.20: In each of 19 through 24: a. Sketch the graph of the given function...
 10.2.21: In each of 19 through 24: a. Sketch the graph of the given function...
 10.2.22: In each of 19 through 24: a. Sketch the graph of the given function...
 10.2.23: In each of 19 through 24: a. Sketch the graph of the given function...
 10.2.24: In each of 19 through 24: a. Sketch the graph of the given function...
 10.2.25: Consider the function f defined in 21, and let em( x) = f ( x) sm( ...
 10.2.26: Consider the function f defined in 24, and let em( x) = f ( x) sm( ...
 10.2.27: Suppose that g is an integrable periodic function with period T . a...
 10.2.28: If f is differentiable and is periodic with period T , show that f ...
 10.2.29: In this problem we indicate certain similarities between threedimen...
Solutions for Chapter 10.2: Fourier Series
Full solutions for Elementary Differential Equations and Boundary Value Problems  11th Edition
ISBN: 9781119256007
Solutions for Chapter 10.2: Fourier Series
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 11. Since 29 problems in chapter 10.2: Fourier Series have been answered, more than 12646 students have viewed full stepbystep solutions from this chapter. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9781119256007. Chapter 10.2: Fourier Series includes 29 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Iterative method.
A sequence of steps intended to approach the desired solution.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).