- 126.96.36.199: What is a path of integration? What did we assume about paths?
- 188.8.131.52: State the definition of a complex line integral from memory.
- 184.108.40.206: What do we mean by saying that complex integration is a linear oper...
- 220.127.116.11: Make a list of integration methods discussed. lllustrate each with ...
- 18.104.22.168: Which integration methods apply to analytic functions only?
- 22.214.171.124: What value do you get if you integrate liz counterclockwise around ...
- 126.96.36.199: Which theorem in this chapter do you regard as most important? Stat...
- 188.8.131.52: What is independence of path? What is the principle of deformation ...
- 184.108.40.206: Do not confuse Cauchy's integral theorem and Cauchy's integral form...
- 220.127.116.11: How can you extend Cauchy's integral theorem to doubly and triply c...
- 18.104.22.168: If integrating fez) over the boundary circles of an annulus D gives...
- 22.214.171.124: Is I fJ(Z) dz I = fel f(::)1 dz? How would you find a bound for the...
- 126.96.36.199: Is Re J fez) dz = J Re fez) dz? Give examples.
- 188.8.131.52: How did we use integral formulas for derivatives in integration?
- 184.108.40.206: What is Liouville's theorem? Give examples. State consequences.
- 220.127.116.11: Integrate by a suitable method:4z3 + 2z from - i to 2 + i along any...
- 18.104.22.168: Integrate by a suitable method:5z - 3/z counterclockwise around the...
- 22.214.171.124: Integrate by a suitable method: :: + liz counterclockwise around I:...
- 126.96.36.199: Integrate by a suitable method:e2z from -2 + 37ri along the straigh...
- 188.8.131.52: Integrate by a suitable method:ez2/(z - 1)2 counterclockwise around...
- 184.108.40.206: Integrate by a suitable method: z1(z2 + 1) clockwise around Iz + il...
- 220.127.116.11: Integrate by a suitable method:Re:: from 0 to 4 and then vertically...
- 18.104.22.168: Integrate by a suitable method:cosh 4z from 0 to 2i along the imagi...
- 22.214.171.124: Integrate by a suitable method:eZ/z over C consisting of 1.::1 = I ...
- 126.96.36.199: Integrate by a suitable method:(sin z)/z clockwise around a circle ...
- 188.8.131.52: Integrate by a suitable method: 1m z counterclockwi~e around /:::1 = r
- 184.108.40.206: Integrate by a suitable method: (Ln z)/(z - 202 counterclockwise ar...
- 220.127.116.11: Integrate by a suitable method:(tan 7r:::)/(z - 1)2 counterclockwis...
- 18.104.22.168: Integrate by a suitable method:+ z clockwise around the unit circle
- 22.214.171.124: Integrate by a suitable method:(z - i)-3(Z3 + sin
Solutions for Chapter 14: Complex Integration
Full solutions for Advanced Engineering Mathematics | 9th Edition
Tv = Av + Vo = linear transformation plus shift.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
A = CTC = (L.J]))(L.J]))T for positive definite A.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
A sequence of steps intended to approach the desired solution.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.
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).
Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. 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.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
Pseudoinverse A+ (Moore-Penrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
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.