 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 26832 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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.

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.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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 •

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

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

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

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

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

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).