 13.13.1: Let SN (x) = 4 N n=1 1 (1)n n3 sin(nx). Construct graphs of SN (x) ...
 13.13.4: In each of 1 through 12, write the Fourier seriesof the function on...
 13.13.19: In each of 1 through 10, write the Fourier cosine and sine series f...
 13.13.32: Let f (x) = 0 for x 0 x for 0 < x . (a) Write the Fourier series of...
 13.13.38: In 1, 2, and 3, let f be periodic of period pIf g is also periodic ...
 13.13.50: In each of 1 through 7, write the complex Fourier series of f , det...
 13.13.57: In each of 1 through 5, graph the function, the fifth partial sum o...
 13.13.2: Let SN (x) = 4 N n=1 1 (1)n n3 sin(nx). Construct graphs of SN (x) ...
 13.13.5: In each of 1 through 12, write the Fourier seriesof the function on...
 13.13.20: In each of 1 through 10, write the Fourier cosine and sine series f...
 13.13.33: Let f (x) = x for 1 x 1. (a) Write the Fourier series for f on [1...
 13.13.39: In 1, 2, and 3, let f be periodic of period pLet be a positive numb...
 13.13.51: In each of 1 through 7, write the complex Fourier series of f , det...
 13.13.58: In each of 1 through 5, graph the function, the fifth partial sum o...
 13.13.3: Let p(x) be a polynomial. Prove that there is no finite sum N n=1 b...
 13.13.6: In each of 1 through 12, write the Fourier seriesof the function on...
 13.13.21: In each of 1 through 10, write the Fourier cosine and sine series f...
 13.13.34: Let f (x) = x sin(x) for x . (a) Write the Fourier series for f on ...
 13.13.40: In 1, 2, and 3, let f be periodic of period pIf f is differentiable...
 13.13.52: In each of 1 through 7, write the complex Fourier series of f , det...
 13.13.59: In each of 1 through 5, graph the function, the fifth partial sum o...
 13.13.7: In each of 1 through 12, write the Fourier seriesof the function on...
 13.13.22: In each of 1 through 10, write the Fourier cosine and sine series f...
 13.13.35: Let f (x) = x 2 for 3 x 3. (a) Write the Fourier series for f on [3...
 13.13.41: In each of 4 through 12, find the phase angle form of the Fourier s...
 13.13.53: In each of 1 through 7, write the complex Fourier series of f , det...
 13.13.60: In each of 1 through 5, graph the function, the fifth partial sum o...
 13.13.8: In each of 1 through 12, write the Fourier seriesof the function on...
 13.13.23: In each of 1 through 10, write the Fourier cosine and sine series f...
 13.13.36: Let f and f be piecewise continuous on [L, L]. Use Bessels inequali...
 13.13.42: In each of 4 through 12, find the phase angle form of the Fourier s...
 13.13.54: In each of 1 through 7, write the complex Fourier series of f , det...
 13.13.61: In each of 1 through 5, graph the function, the fifth partial sum o...
 13.13.9: In each of 1 through 12, write the Fourier seriesof the function on...
 13.13.24: In each of 1 through 10, write the Fourier cosine and sine series f...
 13.13.37: Prove Theorem 13.8 by filling in the details of the following argum...
 13.13.43: In each of 4 through 12, find the phase angle form of the Fourier s...
 13.13.55: In each of 1 through 7, write the complex Fourier series of f , det...
 13.13.62: Let f (t) = 1 for 0 t < 2 1 for 2 t < 0 Plot the fifth partial sum ...
 13.13.10: In each of 1 through 12, write the Fourier seriesof the function on...
 13.13.25: In each of 1 through 10, write the Fourier cosine and sine series f...
 13.13.44: In each of 4 through 12, find the phase angle form of the Fourier s...
 13.13.56: In each of 1 through 7, write the complex Fourier series of f , det...
 13.13.63: Let f (t) = t for 2 t < 0 2 + t for 0 t < 2 Plot the fifth partial ...
 13.13.11: In each of 1 through 12, write the Fourier seriesof the function on...
 13.13.26: In each of 1 through 10, write the Fourier cosine and sine series f...
 13.13.45: In each of 4 through 12, find the phase angle form of the Fourier s...
 13.13.12: In each of 1 through 12, write the Fourier seriesof the function on...
 13.13.27: In each of 1 through 10, write the Fourier cosine and sine series f...
 13.13.46: f has the graph of Figure 13.19. x y 1 1 123 1
 13.13.13: In each of 1 through 12, write the Fourier seriesof the function on...
 13.13.28: In each of 1 through 10, write the Fourier cosine and sine series f...
 13.13.47: f has the graph of Figure 13.20. x y k 2 2 4
 13.13.14: In each of 1 through 12, write the Fourier seriesof the function on...
 13.13.29: Sum the series n=1(1)n /(4n2 1). Hint: Expand sin(x) in a cosine se...
 13.13.48: f has the graph of Figure 13.21. x y 1 2 3 1 1 3
 13.13.15: In each of 1 through 12, write the Fourier seriesof the function on...
 13.13.30: Let f (x) be defined on [L, L]. Prove that f can be written as a su...
 13.13.49: f has the graph of Figure 13.22. y x k 2 1 0 1 23
 13.13.16: In each of 13 through 19, use the convergence theorem to determine ...
 13.13.31: Determine all functions on [L, L] that are both even and odd.
 13.13.17: In each of 13 through 19, use the convergence theorem to determine ...
 13.13.18: In each of 13 through 19, use the convergence theorem to determine ...
Solutions for Chapter 13: Fourier Series
Full solutions for Advanced Engineering Mathematics  7th Edition
ISBN: 9781111427412
Solutions for Chapter 13: Fourier Series
Get Full SolutionsChapter 13: Fourier Series includes 63 full stepbystep solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 7. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781111427412. Since 63 problems in chapter 13: Fourier Series have been answered, more than 7773 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

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.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.