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Solutions for Chapter Chapter 11: THE SCHWARZ-CHRISTOFFEL TRANSFORMATION

Complex Variables and Applications | 9th Edition | ISBN: 9780073383170 | Authors: James Ward Brown

Full solutions for Complex Variables and Applications | 9th Edition

ISBN: 9780073383170

Complex Variables and Applications | 9th Edition | ISBN: 9780073383170 | Authors: James Ward Brown

Solutions for Chapter Chapter 11: THE SCHWARZ-CHRISTOFFEL TRANSFORMATION

Solutions for Chapter Chapter 11
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Textbook: Complex Variables and Applications
Edition: 9
Author: James Ward Brown
ISBN: 9780073383170

This textbook survival guide was created for the textbook: Complex Variables and Applications, edition: 9. Since 22 problems in chapter Chapter 11: THE SCHWARZ-CHRISTOFFEL TRANSFORMATION have been answered, more than 6079 students have viewed full step-by-step solutions from this chapter. Complex Variables and Applications was written by and is associated to the ISBN: 9780073383170. Chapter Chapter 11: THE SCHWARZ-CHRISTOFFEL TRANSFORMATION includes 22 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Key Math Terms and definitions covered in this textbook
  • Big formula for n by n determinants.

    Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.

  • Complex conjugate

    z = a - ib for any complex number z = a + ib. Then zz = Iz12.

  • Determinant IAI = det(A).

    Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

  • Ellipse (or ellipsoid) x T Ax = 1.

    A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad

  • Exponential eAt = I + At + (At)2 12! + ...

    has derivative AeAt; eAt u(O) solves u' = Au.

  • 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.

  • Hermitian matrix A H = AT = A.

    Complex analog a j i = aU of a symmetric matrix.

  • Identity matrix I (or In).

    Diagonal entries = 1, off-diagonal entries = 0.

  • 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.

  • 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.

  • Linear transformation T.

    Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

  • Linearly dependent VI, ... , Vn.

    A combination other than all Ci = 0 gives L Ci Vi = O.

  • Normal matrix.

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

  • Nullspace matrix N.

    The columns of N are the n - r special solutions to As = 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.

  • 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 = M-I AM has the same eigenvalues as A.

  • 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. A-I is also symmetric.

  • Tridiagonal matrix T: tij = 0 if Ii - j I > 1.

    T- 1 has rank 1 above and below diagonal.

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