 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 19110 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.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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

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.

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

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib IIĀ· Condition numbers measure the sensitivity of the output to change in the input.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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.

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.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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

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.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

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

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

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.