 23.23.30: In each of 1 through 6, find a bilinear transformation of the first...
 23.23.39: Upper halfplane: u(x, 0) = f (x)
 23.23.46: In each of 1 through 4, analyze the flow having the given complex p...
 23.23.1: In each of parts (a) through (e), find the image of the rectangle u...
 23.23.31: In each of 1 through 6, find a bilinear transformation of the first...
 23.23.40: Right quarter plane: Re(z) > 0, u(x, 0) = f (x), and u(0, y) = 0.
 23.23.47: In each of 1 through 4, analyze the flow having the given complex p...
 23.23.2: In each of parts (a) through (e), find the image of the rectangle u...
 23.23.32: In each of 1 through 6, find a bilinear transformation of the first...
 23.23.41: The disk z z0 < R if u(x, y) = x y for (x, y) on the boundary
 23.23.48: In each of 1 through 4, analyze the flow having the given complex p...
 23.23.3: In each of parts (a) through (e), find the image of the rectangle u...
 23.23.33: In each of 1 through 6, find a bilinear transformation of the first...
 23.23.42: Right halfplane with boundary condition: u(0, y) = 1 for 1 x 1 0 f...
 23.23.49: In each of 1 through 4, analyze the flow having the given complex p...
 23.23.4: Determine the image of the sector /4 5/4 under the mapping w = z2 ....
 23.23.34: In each of 1 through 6, find a bilinear transformation of the first...
 23.23.43: The unit disk if u(x, y) = x y for (x, y) on the boundary circle
 23.23.50: f (z) = KLog(z z0) with K as a nonzero real constant and z0 as a co...
 23.23.5: Determine the image of the sector /6 /3 under the mapping w = z3 . ...
 23.23.35: In each of 1 through 6, find a bilinear transformation of the first...
 23.23.44: The unit disk with u(ei ) = 1 for 0 /4 0 for /4 << 2
 23.23.51: Let f (z) = KLogz a z b with K as a nonzero real numbers and a and ...
 23.23.6: Show that the mapping w = 1 2 z + 1 z maps the circle z =r onto a...
 23.23.36: Find a conformal mapping of the upper halfplane onto the wedge 9 <...
 23.23.45: Solve the Dirichlet problem for the strip 1 < Im(z) < 1, Re(z) > 0 ...
 23.23.52: Let f (z) = K z + 1 z with K as a nonzero real number. Sketch some ...
 23.23.7: Show that the mapping of maps a halfline = k onto a hyperbola with...
 23.23.37: Let w = f (z) = log(z) be defined by restricting the argument of z ...
 23.23.53: Let f (z) = m ik 2 Logz a z b with m and k as nonzero, real numbers...
 23.23.8: Let D consist of all z in the rectangle having vertices i and i, wi...
 23.23.38: Show that the SchwarzChristoffel transformation f (z) = 2i z 0 ( +...
 23.23.54: Analyze the flow having complex potential f (z) = K z + 1 z + ib 2 ...
 23.23.9: Determine the image of D under the mapping w=2z2 . Sketch this image
 23.23.55: Analyze the flow having potential f (z) = iKa 3Log 2z ia3 2z + ia3 ...
 23.23.10: Determine the image of the infinite strip 0 Im 2 under the mapping ...
 23.23.56: Use Blasiuss theorem to show that the force per unit width on the c...
 23.23.11: In each of 11 through 16, find the image of the given circle or lin...
 23.23.12: In each of 11 through 16, find the image of the given circle or lin...
 23.23.13: In each of 11 through 16, find the image of the given circle or lin...
 23.23.14: In each of 11 through 16, find the image of the given circle or lin...
 23.23.15: In each of 11 through 16, find the image of the given circle or lin...
 23.23.16: In each of 11 through 16, find the image of the given circle or lin...
 23.23.17: In each of 17 through 21, find a bilinear transformation taking the...
 23.23.18: In each of 17 through 21, find a bilinear transformation taking the...
 23.23.19: In each of 17 through 21, find a bilinear transformation taking the...
 23.23.20: In each of 17 through 21, find a bilinear transformation taking the...
 23.23.21: In each of 17 through 21, find a bilinear transformation taking the...
 23.23.22: Prove that the composition of two conformal mappings is conformal.
 23.23.23: Show that the mapping w =T (z)=z is not conformal
 23.23.24: Suppose T is a bilinear transformation that is not the identity map...
 23.23.25: A point z0 is a fixed point of a bilinear transformation T if T (z0...
 23.23.26: Let T and S be bilinear mappings that agree at three points. Show t...
 23.23.27: Define the cross ratio of four complex numbers z1,z2,z3, and z4 to ...
 23.23.28: Show that the cross ratio [z1,z2,z3,z4] is the image of z1 under th...
 23.23.29: Show that a cross ratio [z1,z2,z3,z4] is real if and only if the zj...
Solutions for Chapter 23: Conformal Mappings and Applications
Full solutions for Advanced Engineering Mathematics  7th Edition
ISBN: 9781111427412
Solutions for Chapter 23: Conformal Mappings and Applications
Get Full SolutionsSince 56 problems in chapter 23: Conformal Mappings and Applications have been answered, more than 26971 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 7. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781111427412. Chapter 23: Conformal Mappings and Applications includes 56 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Block matrix.
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.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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.

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

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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