 17.5.1: In 1 and 2, the given function is analytic for all z. Show that the...
 17.5.2: In 1 and 2, the given function is analytic for all z. Show that the...
 17.5.3: In 38, show that the given function is not analytic at any point.
 17.5.4: In 38, show that the given function is not analytic at any point.
 17.5.5: In 38, show that the given function is not analytic at any point.
 17.5.6: In 38, show that the given function is not analytic at any point.
 17.5.7: In 38, show that the given function is not analytic at any point.
 17.5.8: In 38, show that the given function is not analytic at any point.
 17.5.9: In 914, use Theorem 17.5.2 to show that the given function is analy...
 17.5.10: In 914, use Theorem 17.5.2 to show that the given function is analy...
 17.5.11: In 914, use Theorem 17.5.2 to show that the given function is analy...
 17.5.12: In 914, use Theorem 17.5.2 to show that the given function is analy...
 17.5.13: In 914, use Theorem 17.5.2 to show that the given function is analy...
 17.5.14: In 914, use Theorem 17.5.2 to show that the given function is analy...
 17.5.15: In 15 and 16, find real constants a, b, c, and d so that the given ...
 17.5.16: In 15 and 16, find real constants a, b, c, and d so that the given ...
 17.5.17: In 1720, show that the given function is not analytic at any point ...
 17.5.18: In 1720, show that the given function is not analytic at any point ...
 17.5.19: In 1720, show that the given function is not analytic at any point ...
 17.5.20: In 1720, show that the given function is not analytic at any point ...
 17.5.21: Use (8) to find the derivative of the function in 9.
 17.5.22: Use (8) to find the derivative of the function in 11.
 17.5.23: In 2328, verify that the given function u is harmonic. Find v, the ...
 17.5.24: In 2328, verify that the given function u is harmonic. Find v, the ...
 17.5.25: In 2328, verify that the given function u is harmonic. Find v, the ...
 17.5.26: In 2328, verify that the given function u is harmonic. Find v, the ...
 17.5.27: In 2328, verify that the given function u is harmonic. Find v, the ...
 17.5.28: In 2328, verify that the given function u is harmonic. Find v, the ...
 17.5.29: Sketch the level curves u(x, y) c1 and v(x, y) c2 of the analytic f...
 17.5.30: Consider the function f (z) 1/z. Describe the level curves.
 17.5.31: Consider the function f (z) z 1/z. Describe the level curve v(x, y) 0.
 17.5.32: Suppose u and v are the harmonic functions forming the real and ima...
Solutions for Chapter 17.5: CauchyRiemann Equations
Full solutions for Advanced Engineering Mathematics  6th Edition
ISBN: 9781284105902
Solutions for Chapter 17.5: CauchyRiemann Equations
Get Full SolutionsSince 32 problems in chapter 17.5: CauchyRiemann Equations have been answered, more than 36549 students have viewed full stepbystep solutions from this chapter. Chapter 17.5: CauchyRiemann Equations includes 32 full stepbystep solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics , edition: 6. This expansive textbook survival guide covers the following chapters and their solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781284105902.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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).

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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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.

Iterative method.
A sequence of steps intended to approach the desired solution.

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.

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

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Outer product uv T
= column times row = rank one matrix.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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.

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.

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

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.