 12.4.1: In 16, find the complex Fourier series of f on the given interval.
 12.4.2: In 16, find the complex Fourier series of f on the given interval.
 12.4.3: In 16, find the complex Fourier series of f on the given interval.
 12.4.4: In 16, find the complex Fourier series of f on the given interval.
 12.4.5: In 16, find the complex Fourier series of f on the given interval.
 12.4.6: In 16, find the complex Fourier series of f on the given interval.
 12.4.7: Find the frequency spectrum of the periodic wave that is the period...
 12.4.8: Find the frequency spectrum of the periodic wave that is the period...
 12.4.9: In 9 and 10, sketch the given periodic wave. Find the frequency spe...
 12.4.10: In 9 and 10, sketch the given periodic wave. Find the frequency spe...
 12.4.11: (a) Show that an cn cn and bn i(cn cn). (b) Use the results in part...
 12.4.12: The function f in is odd. Use the complex Fourier series to obtain ...
Solutions for Chapter 12.4: Complex Fourier Series
Full solutions for Advanced Engineering Mathematics  6th Edition
ISBN: 9781284105902
Solutions for Chapter 12.4: Complex Fourier Series
Get Full SolutionsThis textbook survival guide was created for the textbook: Advanced Engineering Mathematics , edition: 6. This expansive textbook survival guide covers the following chapters and their solutions. Since 12 problems in chapter 12.4: Complex Fourier Series have been answered, more than 36574 students have viewed full stepbystep solutions from this chapter. Chapter 12.4: Complex Fourier Series includes 12 full stepbystep solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781284105902.

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

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

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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.

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 •

Outer product uv T
= column times row = rank one matrix.

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).

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Solvable system Ax = b.
The right side b is in the column space of A.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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.

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