 11.11.1.139: The functions f(x) x2 1 and g(x) x5 are orthogonal on the interval ...
 11.11.1.140: The product of an odd function f with an odd function g is
 11.11.1.141: To expand f(x) x 1, p x p, in an appropriate trigonometric series, ...
 11.11.1.142: y 0 is never an eigenfunction of a SturmLiouville problem. ___
 11.11.1.143: 0 is never an eigenvalue of a SturmLiouville problem. _
 11.11.1.144: If the function is expanded in a Fourier series, the series will co...
 11.11.1.145: Suppose the function f(x) x2 1, 0 x 3, is expanded in a Fourier ser...
 11.11.1.146: What is the corresponding eigenfunction for the boundaryvalue probl...
 11.11.1.147: Chebyshevs differential equation has a polynomial solution for n 0,...
 11.11.1.148: The set of Legendre polynomials {Pn(x)}, where P0(x) 1, P1(x) x, . ...
 11.11.1.149: Without doing any work, explain why the cosine series of is the fin...
 11.11.1.150: (a) Show that the set is orthogonal on the interval [0, L]. (b) Fin...
 11.11.1.151: Expand in a Fourier series.
 11.11.1.152: Expand f(x) 2x2 1, 1 x 1 in a Fourier series.
 11.11.1.153: Expand (a) in a cosine series (b) in a Fourier series
 11.11.1.154: In 13, 14, and 15, sketch the periodic extension of f to which each...
 11.11.1.155: Discuss: Which of the two Fourier series of f in converges to on th...
 11.11.1.156: Consider the portion of the periodic function f shown in Figure 11....
 11.11.1.157: Find the eigenvalues and eigenfunctions of the boundaryvalue problem
 11.11.1.158: Give an orthogonality relation for the eigenfunctions in 19
 11.11.1.159: Expand , in a FourierBessel series, using Bessel functions of orde...
 11.11.1.160: Expand , in a FourierLegendre series.
 11.11.1.161: Suppose the function y f(x) is defined on the interval . (a) Verify...
 11.11.1.162: The function f(x) ex is neither even or odd. Use to write f as the ...
 11.11.1.163: Suppose that f is an integrable 2pperiodic function. Prove that fo...
Solutions for Chapter 11: Fourier Series
Full solutions for Differential Equations with BoundaryValue Problems,  8th Edition
ISBN: 9781111827069
Solutions for Chapter 11: Fourier Series
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 with BoundaryValue Problems,, edition: 8. Chapter 11: Fourier Series includes 25 full stepbystep solutions. Differential Equations with BoundaryValue Problems, was written by and is associated to the ISBN: 9781111827069. Since 25 problems in chapter 11: Fourier Series have been answered, more than 22640 students have viewed full stepbystep solutions from this chapter.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

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