 21.21.1: In each of 1 through 6, find the radius of convergence and open dis...
 21.21.26: In each of 1 through 10, write the Laurent expansion of f (z) in an...
 21.21.2: In each of 1 through 6, find the radius of convergence and open dis...
 21.21.27: In each of 1 through 10, write the Laurent expansion of f (z) in an...
 21.21.3: In each of 1 through 6, find the radius of convergence and open dis...
 21.21.28: In each of 1 through 10, write the Laurent expansion of f (z) in an...
 21.21.4: In each of 1 through 6, find the radius of convergence and open dis...
 21.21.29: In each of 1 through 10, write the Laurent expansion of f (z) in an...
 21.21.5: In each of 1 through 6, find the radius of convergence and open dis...
 21.21.30: In each of 1 through 10, write the Laurent expansion of f (z) in an...
 21.21.6: In each of 1 through 6, find the radius of convergence and open dis...
 21.21.31: In each of 1 through 10, write the Laurent expansion of f (z) in an...
 21.21.7: Is it possible for n=0 cn (z 2i)n to converge at 0 and diverge i?
 21.21.32: In each of 1 through 10, write the Laurent expansion of f (z) in an...
 21.21.8: Is it possible for n=0 cn (z 4 + 2i)n to converge at i and diverge ...
 21.21.33: In each of 1 through 10, write the Laurent expansion of f (z) in an...
 21.21.9: In each of 9 through 14, find the Taylor expansion of the function ...
 21.21.34: In each of 1 through 10, write the Laurent expansion of f (z) in an...
 21.21.10: In each of 9 through 14, find the Taylor expansion of the function ...
 21.21.35: In each of 1 through 10, write the Laurent expansion of f (z) in an...
 21.21.11: In each of 9 through 14, find the Taylor expansion of the function ...
 21.21.36: Fill in the details of the following proof of the Laurent expansion...
 21.21.12: In each of 9 through 14, find the Taylor expansion of the function ...
 21.21.37: Fill in the details of the following proof of the uniqueness of Lau...
 21.21.13: In each of 9 through 14, find the Taylor expansion of the function ...
 21.21.14: In each of 9 through 14, find the Taylor expansion of the function ...
 21.21.15: Suppose f is differentiable in an open disk about 0 and satisfies f...
 21.21.16: Find the first seven terms of the Maclaurin expansion of f (z) = si...
 21.21.17: Show that n=0 1 (n!)2 = 1 2 2 0 e2z cos( )d. Hint: Show that zn n! ...
 21.21.18: In each of 18 through 24, determine the order of the zero of the fu...
 21.21.19: In each of 18 through 24, determine the order of the zero of the fu...
 21.21.20: In each of 18 through 24, determine the order of the zero of the fu...
 21.21.21: In each of 18 through 24, determine the order of the zero of the fu...
 21.21.22: In each of 18 through 24, determine the order of the zero of the fu...
 21.21.23: In each of 18 through 24, determine the order of the zero of the fu...
 21.21.24: In each of 18 through 24, determine the order of the zero of the fu...
 21.21.25: Suppose f (z) = n=0 an (z z0) n = n=0 bn (z z0) n in some open disk...
Solutions for Chapter 21: Complex Integration
Full solutions for Advanced Engineering Mathematics  7th Edition
ISBN: 9781111427412
Solutions for Chapter 21: Complex Integration
Get Full SolutionsSince 37 problems in chapter 21: Complex Integration have been answered, more than 7743 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 7. Chapter 21: Complex Integration includes 37 full stepbystep solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781111427412.

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

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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.

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

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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).

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.

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.

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

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.

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

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.