 14.4.14.1.86: Imegrate counterclockwise around the circle Izl = 2. (n is a positi...
 14.4.14.1.87: Imegrate counterclockwise around the circle Izl = 2. (n is a positi...
 14.4.14.1.88: Imegrate counterclockwise around the circle Izl = 2. (n is a positi...
 14.4.14.1.89: Imegrate counterclockwise around the circle Izl = 2. (n is a positi...
 14.4.14.1.90: Imegrate counterclockwise around the circle Izl = 2. (n is a positi...
 14.4.14.1.91: Imegrate counterclockwise around the circle Izl = 2. (n is a positi...
 14.4.14.1.92: Imegrate counterclockwise around the circle Izl = 2. (n is a positi...
 14.4.14.1.93: Imegrate counterclockwise around the circle Izl = 2. (n is a positi...
 14.4.14.1.94: Integrate around C. Show the details.
 14.4.14.1.95: Integrate around C. Show the details.
 14.4.14.1.96: Integrate around C. Show the details.
 14.4.14.1.97: Integrate around C. Show the details.
 14.4.14.1.98: Integrate around C. Show the details.
 14.4.14.1.99: TEAM PROJECT. Theory on Growth (a) Growth of entire functions. If f...
 14.4.14.1.100: (Proof of Theorem 1) Complete the proof of Theorem 1 by performing ...
Solutions for Chapter 14.4: Derivatives of Analytic Functions
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter 14.4: Derivatives of Analytic Functions
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 15 problems in chapter 14.4: Derivatives of Analytic Functions have been answered, more than 46330 students have viewed full stepbystep solutions from this chapter. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. Chapter 14.4: Derivatives of Analytic Functions includes 15 full stepbystep solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9.

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.

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

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 II· Condition numbers measure the sensitivity of the output to change in the input.

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.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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).

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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.

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.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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

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.

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!

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