 22.22.1: In each of 1 through 12, determine all singularitiesof the function...
 22.22.14: In each of 1 through 16, use the residue theorem to evaluate the in...
 22.22.36: In each of 1 through 10, evaluate the integral. Wherever they appea...
 22.22.54: In each of 1 through 10, use Theorem 22.7 to find the inverse Lapla...
 22.22.2: In each of 1 through 12, determine all singularitiesof the function...
 22.22.15: In each of 1 through 16, use the residue theorem to evaluate the in...
 22.22.37: In each of 1 through 10, evaluate the integral. Wherever they appea...
 22.22.55: In each of 1 through 10, use Theorem 22.7 to find the inverse Lapla...
 22.22.3: In each of 1 through 12, determine all singularitiesof the function...
 22.22.16: In each of 1 through 16, use the residue theorem to evaluate the in...
 22.22.38: In each of 1 through 10, evaluate the integral. Wherever they appea...
 22.22.56: In each of 1 through 10, use Theorem 22.7 to find the inverse Lapla...
 22.22.4: In each of 1 through 12, determine all singularitiesof the function...
 22.22.17: In each of 1 through 16, use the residue theorem to evaluate the in...
 22.22.39: In each of 1 through 10, evaluate the integral. Wherever they appea...
 22.22.57: In each of 1 through 10, use Theorem 22.7 to find the inverse Lapla...
 22.22.5: In each of 1 through 12, determine all singularitiesof the function...
 22.22.18: In each of 1 through 16, use the residue theorem to evaluate the in...
 22.22.40: In each of 1 through 10, evaluate the integral. Wherever they appea...
 22.22.58: In each of 1 through 10, use Theorem 22.7 to find the inverse Lapla...
 22.22.6: In each of 1 through 12, determine all singularitiesof the function...
 22.22.19: In each of 1 through 16, use the residue theorem to evaluate the in...
 22.22.41: In each of 1 through 10, evaluate the integral. Wherever they appea...
 22.22.59: In each of 1 through 10, use Theorem 22.7 to find the inverse Lapla...
 22.22.7: In each of 1 through 12, determine all singularitiesof the function...
 22.22.20: In each of 1 through 16, use the residue theorem to evaluate the in...
 22.22.42: In each of 1 through 10, evaluate the integral. Wherever they appea...
 22.22.60: In each of 1 through 10, use Theorem 22.7 to find the inverse Lapla...
 22.22.8: In each of 1 through 12, determine all singularitiesof the function...
 22.22.21: In each of 1 through 16, use the residue theorem to evaluate the in...
 22.22.43: In each of 1 through 10, evaluate the integral. Wherever they appea...
 22.22.61: In each of 1 through 10, use Theorem 22.7 to find the inverse Lapla...
 22.22.9: In each of 1 through 12, determine all singularitiesof the function...
 22.22.22: In each of 1 through 16, use the residue theorem to evaluate the in...
 22.22.44: In each of 1 through 10, evaluate the integral. Wherever they appea...
 22.22.62: In each of 1 through 10, use Theorem 22.7 to find the inverse Lapla...
 22.22.10: In each of 1 through 12, determine all singularitiesof the function...
 22.22.23: In each of 1 through 16, use the residue theorem to evaluate the in...
 22.22.45: In each of 1 through 10, evaluate the integral. Wherever they appea...
 22.22.63: In each of 1 through 10, use Theorem 22.7 to find the inverse Lapla...
 22.22.11: In each of 1 through 12, determine all singularitiesof the function...
 22.22.24: In each of 1 through 16, use the residue theorem to evaluate the in...
 22.22.46: Show that cos(x) x 2 + 1 dx = e.
 22.22.12: In each of 1 through 12, determine all singularitiesof the function...
 22.22.25: In each of 1 through 16, use the residue theorem to evaluate the in...
 22.22.47: Show that cos(x) x 2 + 1 dx = e.
 22.22.13: Let f be differentiable at z0 and f (z0) =0. Let g have a pole of o...
 22.22.26: In each of 1 through 16, use the residue theorem to evaluate the in...
 22.22.48: Let =. Show that 2 0 1 2 cos2( ) + 2 sin2 ( ) d = 2
 22.22.27: In each of 1 through 16, use the residue theorem to evaluate the in...
 22.22.49: Show that /2 0 1 + sin2 ( ) d = 2 (1 + ).
 22.22.28: In each of 1 through 16, use the residue theorem to evaluate the in...
 22.22.50: Show that 0 ex2 cos(2x) dx = 2 e2 . Hint: Integrate ez2 about the r...
 22.22.29: In each of 1 through 16, use the residue theorem to evaluate the in...
 22.22.51: Derive Fresnels integrals: 0 cos(x 2 ) dx = 0 sin(x 2 ) dx = 1 2 2 ...
 22.22.30: Let h and g be differentiable at z0 and g(z0) = 0. Suppose h has a ...
 22.22.52: Let and be positive numbers. Show that 0 x sin(x) x 4 + 4 dx = 22 e...
 22.22.31: Suppose f is differentiable at points on a closed path and at all p...
 22.22.53: Let 0 <<. Show that 0 1 ( + cos( ))2 d = (2 2)3/2
 22.22.32: Evaluate z 2 + z2 dz with as the circle z = 2 first by using the ...
 22.22.33: Evaluate tan(z) dz with the circle z = first by using the residue...
 22.22.34: Evaluate z + 1 z2 + 2z + 4 dz with the circle z = 2, first by usi...
 22.22.35: Let p(z) = (z z1)(z z2)(z zn ) with z1, ,zn distinct complex number...
Solutions for Chapter 22: The Residue Theorem
Full solutions for Advanced Engineering Mathematics  7th Edition
ISBN: 9781111427412
Solutions for Chapter 22: The Residue Theorem
Get Full SolutionsChapter 22: The Residue Theorem includes 63 full stepbystep 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 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.

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 SI 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 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!).

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

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

lAII = 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 eiO , 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.