 Chapter Chapter 1: COMPLEX NUMBERS
 Chapter Chapter 10: APPLICATIONS OF CONFORMAL MAPPING
 Chapter Chapter 11: THE SCHWARZCHRISTOFFEL TRANSFORMATION
 Chapter Chapter 12: INTEGRAL FORMULAS OF THE POISSON TYPE
 Chapter Chapter 2: ANALYTIC FUNCTIONS
 Chapter Chapter 3: Elementary Functions
 Chapter Chapter 4: INTEGRALS
 Chapter Chapter 5: SERIES
 Chapter Chapter 6: RESIDUES AND POLES
 Chapter Chapter 7: APPLICATIONS OF RESIDUES
 Chapter Chapter 8: MAPPING BY ELEMENTARY FUNCTIONS
 Chapter Chapter 9: CONFORMAL MAPPING
Complex Variables and Applications 9th Edition  Solutions by Chapter
Full solutions for Complex Variables and Applications  9th Edition
ISBN: 9780073383170
Complex Variables and Applications  9th Edition  Solutions by Chapter
Get Full SolutionsThis expansive textbook survival guide covers the following chapters: 12. This textbook survival guide was created for the textbook: Complex Variables and Applications, edition: 9. Since problems from 12 chapters in Complex Variables and Applications have been answered, more than 1989 students have viewed full stepbystep answer. Complex Variables and Applications was written by Sieva Kozinsky and is associated to the ISBN: 9780073383170. The full stepbystep solution to problem in Complex Variables and Applications were answered by Sieva Kozinsky, our top Math solution expert on 12/23/17, 04:39PM.

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

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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

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

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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.

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.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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!

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.
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