 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 7710 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.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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).

Column space C (A) =
space of all combinations of the columns of A.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

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

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).

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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

Pseudoinverse A+ (MoorePenrose 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).

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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

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