- 126.96.36.199.147: Let f(x) = - I if 0 < x < L f(x) = 1 if 1 < x < 2. f(x) = 0 if x > ...
- 188.8.131.52.148: Let f(x) = x if 0 < x < k, f(x) = 0 if x > k. Find Ic(w),
- 184.108.40.206.149: Derive formula 3 in Table 1 of Sec. 11.10 by integration.
- 220.127.116.11.150: Find the inverse Fourier cosine transform f(x) from the answer to P...
- 18.104.22.168.151: Obtain 9';:-1(1/(1 + w2 )) from Prob. 3 in Sec. 11.7.
- 22.214.171.124.152: Obtain 9';:-I(e-W ) by integration.
- 126.96.36.199.153: Find 9'c(1 - X2 )-1 cos (7TX/2. Hint. Use Prob. 5 in Sec. 11.7.
- 188.8.131.52.154: Let f(x) = x 2 if 0 < x < I and 0 if x> 1. Find 9'cCf).
- 184.108.40.206.155: Does the Fourier cosine transform of X-I sin x exist? Of X-I cos x?...
- 220.127.116.11.156: f(x) = 1 (0 < x < (0) has no Fourier cosine or sine transform. Give...
- 18.104.22.168.157: Find 9's(e-"'-X) by integration.
- 22.214.171.124.158: Find the answer to Prob. 11 from (9b).
- 126.96.36.199.159: Obtain formula 8 in Table II of Sec. 11.1 I from (8b) and a suitabl...
- 188.8.131.52.160: Let f(x) = sinx if 0 < x < 7T and 0 if x> 7T. Find 9's(f). Compare ...
- 184.108.40.206.161: In Table II of Sec. 11.10 obtain formula 2 from formula 4, using r@...
- 220.127.116.11.162: Show that 9'sCx- 1I2) = w- 1I2 by setting wx = t 2 and using S(oo) ...
- 18.104.22.168.163: Obtain 9'sCe-ax) from (8a) and formula 3 in Table I of Sec. 11.10
- 22.214.171.124.164: Show that 9's(x-3/2) = 2w1/2 Hint. Set wx = t 2 , integrate by part...
- 126.96.36.199.165: (Scale change) Using the notation of (5), show that f(ax) has the F...
- 188.8.131.52.166: WRITING PROJECT. Obtaining Fourier Cosine and Sine Transforms. Writ...
Solutions for Chapter 11.8: Fourier Cosine and Sine Transforms
Full solutions for Advanced Engineering Mathematics | 9th Edition
Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Free variable Xi.
Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).
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.
Invert A by row operations on [A I] to reach [I A-I].
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
A sequence of steps intended to approach the desired solution.
lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.
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.
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).
Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).