 17.4.1: In 16, find the image of the given line under the mapping f (z) z 2 .
 17.4.2: In 16, find the image of the given line under the mapping f (z) z 2 .
 17.4.3: In 16, find the image of the given line under the mapping f (z) z 2 .
 17.4.4: In 16, find the image of the given line under the mapping f (z) z 2 .
 17.4.5: In 16, find the image of the given line under the mapping f (z) z 2 .
 17.4.6: In 16, find the image of the given line under the mapping f (z) z 2 .
 17.4.7: In 714, express the given function in the form f (z) u iv.
 17.4.8: In 714, express the given function in the form f (z) u iv.
 17.4.9: In 714, express the given function in the form f (z) u iv.
 17.4.10: In 714, express the given function in the form f (z) u iv.
 17.4.11: In 714, express the given function in the form f (z) u iv.
 17.4.12: In 714, express the given function in the form f (z) u iv.
 17.4.13: In 714, express the given function in the form f (z) u iv.
 17.4.14: In 714, express the given function in the form f (z) u iv.
 17.4.15: In 1518, evaluate the given function at the indicated points.
 17.4.16: In 1518, evaluate the given function at the indicated points.
 17.4.17: In 1518, evaluate the given function at the indicated points.
 17.4.18: In 1518, evaluate the given function at the indicated points.
 17.4.19: In 1922, the given limit exists. Find its value.
 17.4.20: In 1922, the given limit exists. Find its value.
 17.4.21: In 1922, the given limit exists. Find its value.
 17.4.22: In 1922, the given limit exists. Find its value.
 17.4.23: In 23 and 24, show that the given limit does not exist.
 17.4.24: In 23 and 24, show that the given limit does not exist.
 17.4.25: In 25 and 26, use (3) to obtain the indicated derivative of the giv...
 17.4.26: In 25 and 26, use (3) to obtain the indicated derivative of the giv...
 17.4.27: In 2734, use (4)(8) to find the derivative f (z) for the given func...
 17.4.28: In 2734, use (4)(8) to find the derivative f (z) for the given func...
 17.4.29: In 2734, use (4)(8) to find the derivative f (z) for the given func...
 17.4.30: In 2734, use (4)(8) to find the derivative f (z) for the given func...
 17.4.31: In 2734, use (4)(8) to find the derivative f (z) for the given func...
 17.4.32: In 2734, use (4)(8) to find the derivative f (z) for the given func...
 17.4.33: In 2734, use (4)(8) to find the derivative f (z) for the given func...
 17.4.34: In 2734, use (4)(8) to find the derivative f (z) for the given func...
 17.4.35: In 3538, give the points at which the given function will not be an...
 17.4.36: In 3538, give the points at which the given function will not be an...
 17.4.37: In 3538, give the points at which the given function will not be an...
 17.4.38: In 3538, give the points at which the given function will not be an...
 17.4.39: Show that the function f (z) z is nowhere differentiable.
 17.4.40: The function f (z) z 2 is continuous throughout the entire comple...
 17.4.41: In 4144, find the streamlines of the flow associated with the given...
 17.4.42: In 4144, find the streamlines of the flow associated with the given...
 17.4.43: In 4144, find the streamlines of the flow associated with the given...
 17.4.44: In 4144, find the streamlines of the flow associated with the given...
 17.4.45: In 45 and 46, use a graphics calculator or computer to obtain the i...
 17.4.46: In 45 and 46, use a graphics calculator or computer to obtain the i...
Solutions for Chapter 17.4: Functions of a Complex Variable
Full solutions for Advanced Engineering Mathematics  6th Edition
ISBN: 9781284105902
Solutions for Chapter 17.4: Functions of a Complex Variable
Get Full SolutionsSince 46 problems in chapter 17.4: Functions of a Complex Variable have been answered, more than 35170 students have viewed full stepbystep solutions from this chapter. Chapter 17.4: Functions of a Complex Variable includes 46 full stepbystep solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781284105902. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics , edition: 6.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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).

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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

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).

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).