 16.16.1.106: Laurent series generalize Taylor senes. Explain the details.
 16.16.1.107: Can a function have several Laurent series with the same center? Ex...
 16.16.1.108: What is the principal part of a Laurent series? Its significance?
 16.16.1.109: What is a pole? An essential singularity,? Give examples.
 16.16.1.110: What is Picard's theorem? Why did it occur in this chapter?
 16.16.1.111: What is the Riemann sphere? The extended complex plane? Tts signifi...
 16.16.1.112: Is elk2 analytic or singular at infinity? cosh;:.? (;:.  4)3? Expl...
 16.16.1.113: What is the residue? Why is it important?
 16.16.1.114: State formulas for residues from memory.
 16.16.1.115: State some further methods for calculating residues.
 16.16.1.116: What is residue integration? To what kind of complex integrals does...
 16.16.1.117: By what idea can we apply residue integration to real integrals fro...
 16.16.1.118: What is a zero of an analytic function? How are zeros classified?
 16.16.1.119: What are improper integrals? Cauchy principal values? Give examples
 16.16.1.120: Can the residue at a singular point be O? At a simple pole'?
 16.16.1.121: What is a meromorphic function? An entire function? Give examples.
 16.16.1.122: Integrate counterclockwise around C. (Show the details.)tan ;:. 4 ...
 16.16.1.123: Integrate counterclockwise around C. (Show the details.)sin 2;:. ~,...
 16.16.1.124: Integrate counterclockwise around C. (Show the details.)2;:. + i ' ...
 16.16.1.125: Integrate counterclockwise around C. (Show the details.)2 _ i;:. + ...
 16.16.1.126: Integrate counterclockwise around C. (Show the details.) c~sh 5z , ...
 16.16.1.127: Integrate counterclockwise around C. (Show the details.)4z3 + 7z , ...
 16.16.1.128: Integrate counterclockwise around C. (Show the details.)cot 8;:., C...
 16.16.1.129: Integrate counterclockwise around C. (Show the details.) 4_2 _ 1 ,C...
 16.16.1.130: Integrate counterclockwise around C. (Show the details.)cos Zz'n ,1...
 16.16.1.131: Integrate counterclockwise around C. (Show the details.)Z2 + 1 1 ;;...
 16.16.1.132: Integrate counterclockwise around C. (Show the details.)15z +9 , C:...
 16.16.1.133: Integrate counterclockwise around C. (Show the details.) _3 _ , C: ...
 16.16.1.134: Evaluate by the methods of this details):7T de29.0 25  24 cos e
 16.16.1.135: Evaluate by the methods of this details):f7T de , k > 1
 16.16.1.136: Evaluate by the methods of this details):{7T de1  ~ sin e
 16.16.1.137: Evaluate by the methods of this details):
 16.16.1.138: Evaluate by the methods of this details):
 16.16.1.139: Evaluate by the methods of this details):
 16.16.1.140: Evaluate by the methods of this details):
 16.16.1.141: Obtain the answer to Prob. 18 in Sec. 16.4 from the present Prob. 35
Solutions for Chapter 16: Laurent Series. Residue Integration
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter 16: Laurent Series. Residue Integration
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. Since 36 problems in chapter 16: Laurent Series. Residue Integration have been answered, more than 48361 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9. Chapter 16: Laurent Series. Residue Integration includes 36 full stepbystep solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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!

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.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Iterative method.
A sequence of steps intended to approach the desired solution.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

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.

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