 14.14.12: In each of 1 through 10, find the Fourier cosine and sine integral ...
 14.14.23: In each of 1 through 15, find the Fourier transform of the function...
 14.14.51: f (t) = et
 14.14.59: In each of 1 through 6, compute D[u](k) for k = 0,1, ,4.u = [cos(j)...
 14.14.75: In each of 1 through 6, a function is given having period p. Comput...
 14.14.81: In each of 1 through 4, make a DFT approximation to the Fourier tra...
 14.14.1: In each of 1 through 10, write the Fourier integral representation ...
 14.14.13: In each of 1 through 10, find the Fourier cosine and sine integral ...
 14.14.24: In each of 1 through 15, find the Fourier transform of the function...
 14.14.52: f (t) = teat with a any positive number
 14.14.60: In each of 1 through 6, compute D[u](k) for k = 0,1, ,4.]u = [ei j]...
 14.14.76: In each of 1 through 6, a function is given having period p. Comput...
 14.14.82: In each of 1 through 4, make a DFT approximation to the Fourier tra...
 14.14.2: In each of 1 through 10, write the Fourier integral representation ...
 14.14.14: In each of 1 through 10, find the Fourier cosine and sine integral ...
 14.14.25: In each of 1 through 15, find the Fourier transform of the function...
 14.14.53: f (t) = cos(t) for 0 t K 0 for t > K with K any positive number.
 14.14.61: In each of 1 through 6, compute D[u](k) for k = 0,1, ,4.1/(j + 1)] ...
 14.14.77: In each of 1 through 6, a function is given having period p. Comput...
 14.14.83: In each of 1 through 4, make a DFT approximation to the Fourier tra...
 14.14.3: In each of 1 through 10, write the Fourier integral representation ...
 14.14.15: In each of 1 through 10, find the Fourier cosine and sine integral ...
 14.14.26: In each of 1 through 15, find the Fourier transform of the function...
 14.14.54: f (t) = 1 for 0 t < K 1 for K t < 2K 0 for t 2K
 14.14.62: In each of 1 through 6, compute D[u](k) for k = 0,1, ,4.[1/(j + 1)2...
 14.14.78: In each of 1 through 6, a function is given having period p. Comput...
 14.14.84: In each of 1 through 4, make a DFT approximation to the Fourier tra...
 14.14.4: In each of 1 through 10, write the Fourier integral representation ...
 14.14.16: In each of 1 through 10, find the Fourier cosine and sine integral ...
 14.14.27: In each of 1 through 15, find the Fourier transform of the function...
 14.14.55: f (t) = et cos(t)
 14.14.63: In each of 1 through 6, compute D[u](k) for k = 0,1, ,4.[j 2 ] 5 j=0
 14.14.79: In each of 1 through 6, a function is given having period p. Comput...
 14.14.5: In each of 1 through 10, write the Fourier integral representation ...
 14.14.17: In each of 1 through 10, find the Fourier cosine and sine integral ...
 14.14.28: In each of 1 through 15, find the Fourier transform of the function...
 14.14.56: f (t) = sinh(t) for K t < 2K 0 for 0 t < K and for t 2K
 14.14.64: In each of 1 through 6, compute D[u](k) for k = 0,1, ,4. [cos(j) si...
 14.14.80: In each of 1 through 6, a function is given having period p. Comput...
 14.14.6: In each of 1 through 10, write the Fourier integral representation ...
 14.14.18: In each of 1 through 10, find the Fourier cosine and sine integral ...
 14.14.29: In each of 1 through 15, find the Fourier transform of the function...
 14.14.57: Show that, under appropriate conditions on f and its derivatives, F...
 14.14.65: In each of 7 through 12, a sequence [Uk ]N k=0 is given. Determine ...
 14.14.7: In each of 1 through 10, write the Fourier integral representation ...
 14.14.19: In each of 1 through 10, find the Fourier cosine and sine integral ...
 14.14.30: In each of 1 through 15, find the Fourier transform of the function...
 14.14.58: Show that, under appropriate conditions on f and its derivatives, F...
 14.14.66: In each of 7 through 12, a sequence [Uk ]N k=0 is given. Determine ...
 14.14.8: In each of 1 through 10, write the Fourier integral representation ...
 14.14.20: In each of 1 through 10, find the Fourier cosine and sine integral ...
 14.14.31: In each of 1 through 15, find the Fourier transform of the function...
 14.14.67: In each of 7 through 12, a sequence [Uk ]N k=0 is given. Determine ...
 14.14.9: In each of 1 through 10, write the Fourier integral representation ...
 14.14.10: In each of 1 through 10, write the Fourier integral representation ...
 14.14.21: In each of 1 through 10, find the Fourier cosine and sine integral ...
 14.14.32: In each of 1 through 15, find the Fourier transform of the function...
 14.14.68: In each of 7 through 12, a sequence [Uk ]N k=0 is given. Determine ...
 14.14.11: Show that the Fourier integral of f (x) can be written lim 1 f (t) ...
 14.14.22: Use the Laplace integrals to compute the Fourier cosine integral of...
 14.14.33: In each of 11 through 15, find the inverse Fourier transform of the...
 14.14.69: In each of 7 through 12, a sequence [Uk ]N k=0 is given. Determine ...
 14.14.34: In each of 11 through 15, find the inverse Fourier transform of the...
 14.14.70: In each of 7 through 12, a sequence [Uk ]N k=0 is given. Determine ...
 14.14.35: In each of 11 through 15, find the inverse Fourier transform of the...
 14.14.71: In each of 13 through 16, compute the first seven complex Fourier c...
 14.14.36: In each of 11 through 15, find the inverse Fourier transform of the...
 14.14.72: In each of 13 through 16, compute the first seven complex Fourier c...
 14.14.37: In each of 11 through 15, find the inverse Fourier transform of the...
 14.14.73: In each of 13 through 16, compute the first seven complex Fourier c...
 14.14.38: In each of 16, 17, and 18, use convolution to find the inverse Four...
 14.14.74: In each of 13 through 16, compute the first seven complex Fourier c...
 14.14.39: In each of 16, 17, and 18, use convolution to find the inverse Four...
 14.14.40: In each of 16, 17, and 18, use convolution to findthe inverse Fouri...
 14.14.41: Prove the following version of Parsevals theorem:  f (t) 2 dt = 1...
 14.14.42: Compute the total energy of the signal f (t) = H(t)e2t .
 14.14.43: Compute the total energy of the signal f (t) = (1/t) sin(3t). Hint:...
 14.14.44: Use the Fourier transform to solve y+ 6y + 5y = (t 3).
 14.14.45: In each of 23 through 28, compute the windowed Fourier transform of...
 14.14.46: In each of 23 through 28, compute the windowed Fourier transform of...
 14.14.47: In each of 23 through 28, compute the windowed Fourier transform of...
 14.14.48: In each of 23 through 28, compute the windowed Fourier transform of...
 14.14.49: In each of 23 through 28, compute the windowed Fourier transform of...
 14.14.50: In each of 23 through 28, compute the windowed Fourier transform of...
Solutions for Chapter 14: Fourier Series
Full solutions for Advanced Engineering Mathematics  7th Edition
ISBN: 9781111427412
Solutions for Chapter 14: Fourier Series
Get Full SolutionsThis textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 7. Chapter 14: Fourier Series includes 84 full stepbystep solutions. Since 84 problems in chapter 14: 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. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781111427412.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Column space C (A) =
space of all combinations of the columns of A.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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.

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.

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.

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

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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.

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.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)ยท(b  Ax) = o.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

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.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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

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