 17.6.1: In 110, express e z in the form a ib.
 17.6.2: In 110, express e z in the form a ib.
 17.6.3: In 110, express e z in the form a ib.
 17.6.4: In 110, express e z in the form a ib.
 17.6.5: In 110, express e z in the form a ib.
 17.6.6: In 110, express e z in the form a ib.
 17.6.7: In 110, express e z in the form a ib.
 17.6.8: In 110, express e z in the form a ib.
 17.6.9: In 110, express e z in the form a ib.
 17.6.10: In 110, express e z in the form a ib.
 17.6.11: In 11 and 12, express the given number in the form a ib.
 17.6.12: In 11 and 12, express the given number in the form a ib.
 17.6.13: In 1316, use Definition 17.6.1 to express the given function in the...
 17.6.14: In 1316, use Definition 17.6.1 to express the given function in the...
 17.6.15: In 1316, use Definition 17.6.1 to express the given function in the...
 17.6.16: In 1316, use Definition 17.6.1 to express the given function in the...
 17.6.17: In 1720, verify the given result.
 17.6.18: In 1720, verify the given result.
 17.6.19: In 1720, verify the given result.
 17.6.20: In 1720, verify the given result.
 17.6.21: Show that f (z) ez 2 is nowhere analytic.
 17.6.22: (a) Use the result in to show that f (z) ez 2 is an entire function...
 17.6.23: In 2328, express ln z in the form a ib.
 17.6.24: In 2328, express ln z in the form a ib.
 17.6.25: In 2328, express ln z in the form a ib.
 17.6.26: In 2328, express ln z in the form a ib.
 17.6.27: In 2328, express ln z in the form a ib.
 17.6.28: In 2328, express ln z in the form a ib.
 17.6.29: In 2934, express Ln z in the form a ib.
 17.6.30: In 2934, express Ln z in the form a ib.
 17.6.31: In 2934, express Ln z in the form a ib.
 17.6.32: In 2934, express Ln z in the form a ib.
 17.6.33: In 2934, express Ln z in the form a ib.
 17.6.34: In 2934, express Ln z in the form a ib.
 17.6.35: In 3538, find all values of z satisfying the given equation.
 17.6.36: In 3538, find all values of z satisfying the given equation.
 17.6.37: In 3538, find all values of z satisfying the given equation.
 17.6.38: In 3538, find all values of z satisfying the given equation.
 17.6.39: In 3942, find all values of the given quantity
 17.6.40: In 3942, find all values of the given quantity
 17.6.41: In 3942, find all values of the given quantity
 17.6.42: In 3942, find all values of the given quantity
 17.6.43: In 43 and 44, find the principal value of the given quantity. Expre...
 17.6.44: In 43 and 44, find the principal value of the given quantity. Expre...
 17.6.45: If z1 i and z2 1 i, verify that Ln(z1z2) Ln z1 Ln z2.
 17.6.46: Find two complex numbers z1 and z2 such that Ln(z1/z2) Ln z1 Ln z2.
 17.6.47: Determine whether the given statement is true. (a) Ln(1 i) 2 2 Ln(1...
 17.6.48: The laws of exponents hold for complex numbers a and b: za z b z ab...
 17.6.49: For complex numbers z satisfying Re(z) 0, show that (7) can be writ...
 17.6.50: The function given in is analytic. (a) Verify that u(x, y) loge(x2 ...
Solutions for Chapter 17.6: Exponential and Logarithmic Functions
Full solutions for Advanced Engineering Mathematics  6th Edition
ISBN: 9781284105902
Solutions for Chapter 17.6: Exponential and Logarithmic Functions
Get Full SolutionsAdvanced Engineering Mathematics was written by and is associated to the ISBN: 9781284105902. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics , edition: 6. Chapter 17.6: Exponential and Logarithmic Functions includes 50 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 50 problems in chapter 17.6: Exponential and Logarithmic Functions have been answered, more than 34627 students have viewed full stepbystep solutions from this chapter.

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

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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.

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.

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.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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).

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).