 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 26970 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.

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.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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].

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

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

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

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

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

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

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).