 16.2.16.1.26: Determine the location and kind of the singularities of the followi...
 16.2.16.1.27: Determine the location and kind of the singularities of the followi...
 16.2.16.1.28: Determine the location and kind of the singularities of the followi...
 16.2.16.1.29: Determine the location and kind of the singularities of the followi...
 16.2.16.1.30: Determine the location and kind of the singularities of the followi...
 16.2.16.1.31: Determine the location and kind of the singularities of the followi...
 16.2.16.1.32: Determine the location and kind of the singularities of the followi...
 16.2.16.1.33: Determine the location and kind of the singularities of the followi...
 16.2.16.1.34: Determine the location and kind of the singularities of the followi...
 16.2.16.1.35: Determine the location and kind of the singularities of the followi...
 16.2.16.1.36: (Essential singularity) Discuss e llz2 in a similar way as e llz is...
 16.2.16.1.37: (Poles) Verify Theorem I for f(:) = :::3  ZI. Prove Theorem 1.
 16.2.16.1.38: Determine the location and order of the zeros. (z + 16i)4
 16.2.16.1.39: Determine the location and order of the zeros.(Z4  16)4
 16.2.16.1.40: Determine the location and order of the zeros.:::3 sin3 7fZ
 16.2.16.1.41: Determine the location and order of the zeros.cosh2
 16.2.16.1.42: Determine the location and order of the zeros.(3z2 + l)e Z
 16.2.16.1.43: Determine the location and order of the zeros. (Z2  1)2(eZ2  L)
 16.2.16.1.44: Determine the location and order of the zeros. (,2 + 4)(eZ  l)2
 16.2.16.1.45: Determine the location and order of the zeros.(sin z  1)3
 16.2.16.1.46: Determine the location and order of the zeros. (1  cos Z)2
 16.2.16.1.47: Determine the location and order of the zeros.
 16.2.16.1.48: (Zeros) If f(:) is analytic and has a zero of order 11 at z = :0' s...
 16.2.16.1.49: TEAM PROJECT. Zeros. la) Derivative. Show that if f(:) has a zero o...
 16.2.16.1.50: (Riemann sphere) Assuming that we let the image of the xaxis be me...
Solutions for Chapter 16.2: Singularities and Zeros. Infinity
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter 16.2: Singularities and Zeros. Infinity
Get Full SolutionsChapter 16.2: Singularities and Zeros. Infinity includes 25 full stepbystep solutions. Since 25 problems in chapter 16.2: Singularities and Zeros. Infinity have been answered, more than 49464 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

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 IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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!).

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Outer product uv T
= column times row = rank one matrix.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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).

Solvable system Ax = b.
The right side b is in the column space of A.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.