 13.6.13.1.132: Prove that cos z, sin z, cosh z, sinh Z are entire functions.
 13.6.13.1.133: Verify by differentiation that Re cos z and 1m sin z are harmonic.
 13.6.13.1.134: Show that cosh z = cosh x cos Y + i sinh x sin ysinh z = sinh x cos...
 13.6.13.1.135: Show that cosh (ZI + Z2) = cosh ZI cosh Z2 + sinh ZI sinh Z2 sinh (...
 13.6.13.1.136: Show that cosh2 Z  sinh2 z. = 1
 13.6.13.1.137: Show that cosh2 Z + sinh2 Z = cosh 2z
 13.6.13.1.138: Function Values. Compute (in the form u + iv)
 13.6.13.1.139: Function Values. Compute (in the form u + iv)
 13.6.13.1.140: Function Values. Compute (in the form u + iv)
 13.6.13.1.141: Function Values. Compute (in the form u + iv)
 13.6.13.1.142: Function Values. Compute (in the form u + iv)
 13.6.13.1.143: Function Values. Compute (in the form u + iv)
 13.6.13.1.144: Function Values. Compute (in the form u + iv)
 13.6.13.1.145: Function Values. Compute (in the form u + iv) sinh (4  3i)
 13.6.13.1.146: Function Values. Compute (in the form u + iv) cosh (4  67Ti)
 13.6.13.1.147: (Real and imaginary parts) Show that sin x cos x Re tan z = =...
 13.6.13.1.148: Equations. Find all solutions of the following equations. cosh z = 0
 13.6.13.1.149: Equations. Find all solutions of the following equations. sin z = 100
 13.6.13.1.150: Equations. Find all solutions of the following equations.cos Z = 2i
 13.6.13.1.151: Equations. Find all solutions of the following equations.cosh z =  1
 13.6.13.1.152: Equations. Find all solutions of the following equations.sinh z = 0
 13.6.13.1.153: Find all z for which (a) cos z, (b) sin z has real values.
 13.6.13.1.154: Equations and Inequalities. Using the definitions, prove: cos z is ...
 13.6.13.1.155: Equations and Inequalities. Using the definitions, prove: Isinh yl ...
 13.6.13.1.156: Equations and Inequalities. Using the definitions, prove:sin ZI cos...
Solutions for Chapter 13.6: Trigonometric and Hyperbolic Functions
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter 13.6: Trigonometric and Hyperbolic Functions
Get Full SolutionsSince 25 problems in chapter 13.6: Trigonometric and Hyperbolic Functions have been answered, more than 48838 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9. Chapter 13.6: Trigonometric and Hyperbolic Functions includes 25 full stepbystep solutions. 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.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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

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.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = 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).

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

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 •

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.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(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.