 14.2.14.1.36: Integrate f(::) counterclockwise around the unit circle. indicating...
 14.2.14.1.37: Integrate f(::) counterclockwise around the unit circle. indicating...
 14.2.14.1.38: Integrate f(::) counterclockwise around the unit circle. indicating...
 14.2.14.1.39: Integrate f(::) counterclockwise around the unit circle. indicating...
 14.2.14.1.40: Integrate f(::) counterclockwise around the unit circle. indicating...
 14.2.14.1.41: Integrate f(::) counterclockwise around the unit circle. indicating...
 14.2.14.1.42: Integrate f(::) counterclockwise around the unit circle. indicating...
 14.2.14.1.43: Integrate f(::) counterclockwise around the unit circle. indicating...
 14.2.14.1.44: Integrate f(::) counterclockwise around the unit circle. indicating...
 14.2.14.1.45: Integrate f(::) counterclockwise around the unit circle. indicating...
 14.2.14.1.46: Integrate f(::) counterclockwise around the unit circle. indicating...
 14.2.14.1.47: (Singularities) Can we conclude in Example 2 that the integral of 1...
 14.2.14.1.48: (Cauchy's integral theorem) Velify Theorem 1 for the integral of ::...
 14.2.14.1.49: (Cauchy's integral theorem) For what contours C will it follow from...
 14.2.14.1.50: (Deformation principle) Can we conclude from Example 4 that the int...
 14.2.14.1.51: (Deformation principle) If the integral of a function fez) over the...
 14.2.14.1.52: (Path independence) Verify Theorem 2 for [he integral of cos:: from...
 14.2.14.1.53: TEAM PROJECT. Cauchy's Integral Theorem. (a) Main Aspects. Each of ...
 14.2.14.1.54: Evaluate (showing the details and using partial fractions if necess...
 14.2.14.1.55: Evaluate (showing the details and using partial fractions if necess...
 14.2.14.1.56: Evaluate (showing the details and using partial fractions if necess...
 14.2.14.1.57: Evaluate (showing the details and using partial fractions if necess...
 14.2.14.1.58: Evaluate (showing the details and using partial fractions if necess...
 14.2.14.1.59: Evaluate (showing the details and using partial fractions if necess...
 14.2.14.1.60: Evaluate (showing the details and using partial fractions if necess...
 14.2.14.1.61: Evaluate (showing the details and using partial fractions if necess...
 14.2.14.1.62: Evaluate (showing the details and using partial fractions if necess...
 14.2.14.1.63: Evaluate (showing the details and using partial fractions if necess...
 14.2.14.1.64: Evaluate (showing the details and using partial fractions if necess...
 14.2.14.1.65: Evaluate (showing the details and using partial fractions if necess...
Solutions for Chapter 14.2: Cauchy's Integral Theorem
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter 14.2: Cauchy's Integral Theorem
Get Full SolutionsThis textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 14.2: Cauchy's Integral Theorem includes 30 full stepbystep solutions. Since 30 problems in chapter 14.2: Cauchy's Integral Theorem have been answered, more than 49079 students have viewed full stepbystep solutions from this chapter.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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 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).

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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

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.

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

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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

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

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

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.