- Chapter 11.11.1: \Ve rcprcsem 1he unil vec1or which is rnngem 10 a smoolh arc C at a...
- Chapter 11.11.2: Obtain inequality (5). Sec. 128. Sugge.\tio11: Let R he larger than...
- Chapter 11.11.5: For an equilateral triangle. k 1 = k 2 = k_, = 2/3. It is convenien...
- Chapter 11.11.7: In transformation (I). Sec. 129. write : 0 = 0. B = 0. and .. . ."1...
- Chapter 11.11.16: \Ve now prcscm a further example of lhc idealized steady How u-cmcd...
- Chapter 11.11.17: To further illuslratc lhc use of lhc Schwarz-Chrisloffel lransfonna...
- Chapter 11.11.18: Use the Schwarz-Christoffel transformation to obtain formally the m...
- Chapter 11.11.3: Use condition (5). Sec. 128. and sufllcient conditions for the exis...
- Chapter 11.11.6: Let us locale the vertices of the rectangle \vhen a > I. As shown i...
- Chapter 11.11.8: Obtain expressions ( 12) in Sec. 129 for the rest of the vertices o...
- Chapter 11.11.19: Explain why the solution of the problem of llow in a channel with a...
- Chapter 11.11.4: According to Sec. 93. the expression I ;, g'(:) N = -- --cl: 2;ri ....
- Chapter 11.11.9: Show that \vhen 0 < <1 < I in expression (8). Sec. 129. the vertice...
- Chapter 11.11.20: Refer to Fig. 29. Appendix 2. As a point ::: moves to the right alo...
- Chapter 11.11.10: Show that the special case 11~=1r(J+1i-1.:2(s-1r-1.:2_\_-1,:2".\' ....
- Chapter 11.11.21: Let T(l1. l') denote the hounded steady-state temperatures in the s...
- Chapter 11.11.11: lUse the Schwarz-Christoffel transformation to arrive at the transf...
- Chapter 11.11.22: Let F( 1r) denote the complex potemi al function for the flow of an...
- Chapter 11.11.12: Refer lo fig. 26 in Appendix 2. As a point::. moves to the right al...
- Chapter 11.11.13: As a point:: moves to the right along that pan of the negative real...
- Chapter 11.11.14: The inverse of the linear fractional transformation i - ::. Z=-- i ...
- Chapter 11.11.15: In the integral of Exercise 8. let the numbers Z i ( j = I. 2 . ......
Solutions for Chapter Chapter 11: THE SCHWARZ-CHRISTOFFEL TRANSFORMATION
Full solutions for Complex Variables and Applications | 9th Edition
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.
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.
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.