 9.2.1: In 1 through 14, the values of a periodic function f .t / in one fu...
 9.2.2: In 1 through 14, the values of a periodic function f .t / in one fu...
 9.2.3: In 1 through 14, the values of a periodic function f .t / in one fu...
 9.2.4: In 1 through 14, the values of a periodic function f .t / in one fu...
 9.2.5: In 1 through 14, the values of a periodic function f .t / in one fu...
 9.2.6: In 1 through 14, the values of a periodic function f .t / in one fu...
 9.2.7: In 1 through 14, the values of a periodic function f .t / in one fu...
 9.2.8: In 1 through 14, the values of a periodic function f .t / in one fu...
 9.2.9: In 1 through 14, the values of a periodic function f .t / in one fu...
 9.2.10: In 1 through 14, the values of a periodic function f .t / in one fu...
 9.2.11: In 1 through 14, the values of a periodic function f .t / in one fu...
 9.2.12: In 1 through 14, the values of a periodic function f .t / in one fu...
 9.2.13: In 1 through 14, the values of a periodic function f .t / in one fu...
 9.2.14: In 1 through 14, the values of a periodic function f .t / in one fu...
 9.2.15: (a) Suppose that f is a function of period 2with f .t / D t2 for 0 ...
 9.2.16: (a) Suppose that f is a function of period 2 such that f .t / D 0 if 1
 9.2.17: (a) Suppose that f is a function of period 2 with f .t / D t for 0
 9.2.18: Derive the Fourier series listed in 18 through 21, and graph the pe...
 9.2.19: Derive the Fourier series listed in 18 through 21, and graph the pe...
 9.2.20: Derive the Fourier series listed in 18 through 21, and graph the pe...
 9.2.21: Derive the Fourier series listed in 18 through 21, and graph the pe...
 9.2.22: Suppose that p.t / is a polynomial of degree n. Show by repeated in...
 9.2.23: Apply the integral formula of to show that Z t 4 cost dt D t 4 sin ...
 9.2.24: Apply the integral formula of to show that Z t 4 cost dt D t 4 sin ...
 9.2.25: (a) Find the Fourier series of the period 2function f with f .t / D...
Solutions for Chapter 9.2: General Fourier Series and Convergence
Full solutions for Differential Equations and Boundary Value Problems: Computing and Modeling  5th Edition
ISBN: 9780321796981
Solutions for Chapter 9.2: General Fourier Series and Convergence
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Differential Equations and Boundary Value Problems: Computing and Modeling, edition: 5. Since 25 problems in chapter 9.2: General Fourier Series and Convergence have been answered, more than 15313 students have viewed full stepbystep solutions from this chapter. Differential Equations and Boundary Value Problems: Computing and Modeling was written by and is associated to the ISBN: 9780321796981. Chapter 9.2: General Fourier Series and Convergence includes 25 full stepbystep solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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

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.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.