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Solutions for Chapter 22: The Residue Theorem

Advanced Engineering Mathematics | 7th Edition | ISBN: 9781111427412 | Authors: Peter V. O'Neill

Full solutions for Advanced Engineering Mathematics | 7th Edition

ISBN: 9781111427412

Advanced Engineering Mathematics | 7th Edition | ISBN: 9781111427412 | Authors: Peter V. O'Neill

Solutions for Chapter 22: The Residue Theorem

Solutions for Chapter 22
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Textbook: Advanced Engineering Mathematics
Edition: 7
Author: Peter V. O'Neill
ISBN: 9781111427412

Chapter 22: The Residue Theorem includes 63 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 63 problems in chapter 22: The Residue Theorem have been answered, more than 7773 students have viewed full step-by-step 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.

Key Math Terms and definitions covered in this textbook
  • Augmented matrix [A b].

    Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

  • Back substitution.

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

  • Cofactor Cij.

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

  • Column space C (A) =

    space of all combinations of the columns of A.

  • Diagonalizable matrix A.

    Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.

  • Factorization

    A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

  • 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 Fn-1c 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!).

  • Hankel matrix H.

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

  • Indefinite matrix.

    A symmetric matrix with eigenvalues of both signs (+ and - ).

  • lA-II = l/lAI and IATI = IAI.

    The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.

  • Multiplicities AM and G M.

    The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

  • 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.

  • Normal matrix.

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

  • Rank r (A)

    = number of pivots = dimension of column space = dimension of row space.

  • Rotation matrix

    R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().

  • Schwarz inequality

    Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

  • 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.

  • Vector addition.

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

  • Volume of box.

    The rows (or the columns) of A generate a box with volume I det(A) I.

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