 15.15.1: In each of 1 through 10, classify the SturmLiouvilleproblem as regu...
 15.15.19: Use the recurrence relation to derive P6(x), P7(x), and P8(x)
 15.15.33: Show that x a J (bxc ) is a solution of y 2a 1 x y + b2 c2 x 2c2 + ...
 15.15.2: In each of 1 through 10, classify the SturmLiouvilleproblem as regu...
 15.15.20: Use Rodriguess formula to derive P2(x) through P5(x)
 15.15.34: In each of 2 through 9, write the general solution of the different...
 15.15.3: In each of 1 through 10, classify the SturmLiouvilleproblem as regu...
 15.15.21: It can be shown that Pn (x) = [n/2] k=0 (1) k (2n 2k)! 2n k!(n k)!(...
 15.15.35: In each of 2 through 9, write the general solution of the different...
 15.15.4: In each of 1 through 10, classify the SturmLiouvilleproblem as regu...
 15.15.22: Put = n(n + 1) into Legendres differential equation, and let y(x) =...
 15.15.36: In each of 2 through 9, write the general solution of the different...
 15.15.5: In each of 1 through 10, classify the SturmLiouvilleproblem as regu...
 15.15.23: The gravitational potential at a point P :(x, y,z) due to a unit ma...
 15.15.37: In each of 2 through 9, write the general solution of the different...
 15.15.6: In each of 1 through 10, classify the SturmLiouvilleproblem as regu...
 15.15.24: Show that n=0 1 2n+1 Pn (1/2) = 1 3 .
 15.15.38: In each of 2 through 9, write the general solution of the different...
 15.15.7: In each of 1 through 10, classify the SturmLiouvilleproblem as regu...
 15.15.25: Let n be a nonnegative integer. Prove that P2n+1(0) = 0 and P2n (0)...
 15.15.39: In each of 2 through 9, write the general solution of the different...
 15.15.8: In each of 1 through 10, classify the SturmLiouvilleproblem as regu...
 15.15.26: Expand each of the following polynomials in a series of Legendre po...
 15.15.40: In each of 2 through 9, write the general solution of the different...
 15.15.9: In each of 1 through 10, classify the SturmLiouvilleproblem as regu...
 15.15.27: In each of 9 through 14, find the first five coef ficients in the ...
 15.15.41: In each of 2 through 9, write the general solution of the different...
 15.15.10: In each of 1 through 10, classify the SturmLiouvilleproblem as regu...
 15.15.28: In each of 9 through 14, find the first five coef ficients in the ...
 15.15.42: Use the change of variables bu = 1 u du dx to transform the differe...
 15.15.11: In each of 11 through 16, find the eigenfunction expansion of the g...
 15.15.29: In each of 9 through 14, find the first five coef ficients in the ...
 15.15.43: In each of 11 through 16, use the given change of variables to tran...
 15.15.12: In each of 11 through 16, find the eigenfunction expansion of the g...
 15.15.30: In each of 9 through 14, find the first five coef ficients in the ...
 15.15.44: In each of 11 through 16, use the given change of variables to tran...
 15.15.13: In each of 11 through 16, find the eigenfunction expansion of the g...
 15.15.31: In each of 9 through 14, find the first five coef ficients in the ...
 15.15.45: In each of 11 through 16, use the given change of variables to tran...
 15.15.14: In each of 11 through 16, find the eigenfunction expansion of the g...
 15.15.32: In each of 9 through 14, find the first five coef ficients in the ...
 15.15.46: In each of 11 through 16, use the given change of variables to tran...
 15.15.15: In each of 11 through 16, find the eigenfunction expansion of the g...
 15.15.47: In each of 11 through 16, use the given change of variables to tran...
 15.15.16: In each of 11 through 16, find the eigenfunction expansion of the g...
 15.15.48: In each of 11 through 16, use the given change of variables to tran...
 15.15.17: Write Bessels inequality for f (x) = x(4 x) for the eigenfunctions ...
 15.15.49: Let be a positive zero of J0. Show that 1 0 J1(x) dx = 1 .
 15.15.18: Write Bessels inequality for f (x)=ex for the eigenfunctions of the...
 15.15.50: Let u(x) = J0(x) and v(x) = J0(x), with and positive constants. (a)...
 15.15.51: Show that, for any positive integer n, x n Jn1(x) dx = x n Jn (x) a...
 15.15.52: Show that, for any positive integer n and any nonzero number , x n ...
 15.15.53: For any nonzero number, and n and k nonnegative integers, define In...
 15.15.54: Use the fact that J1/2(xt) = 2 xt cos(xt) to show that, if n is a p...
 15.15.55: Show that, if m is a positive integer, then Jm (x) = x m 2m1 (m + 1...
 15.15.56: In each of 24 through 29, find (approximately) the first five terms...
 15.15.57: In each of 24 through 29, find (approximately) the first five terms...
 15.15.58: In each of 24 through 29, find (approximately) the first five terms...
 15.15.59: In each of 24 through 29, find (approximately) the first five terms...
 15.15.60: In each of 24 through 29, find (approximately) the first five terms...
 15.15.61: In each of 24 through 29, find (approximately) the first five terms...
 15.15.62: For each of 30 through 35, find (approximately) the first five term...
 15.15.63: For each of 30 through 35, find (approximately) the first five term...
 15.15.64: For each of 30 through 35, find (approximately) the first five term...
 15.15.65: For each of 30 through 35, find (approximately) the first five term...
 15.15.66: For each of 30 through 35, find (approximately) the first five term...
 15.15.67: For each of 30 through 35, find (approximately) the first five term...
 15.15.68: Show that (1/2) = , using the fact from statistics that 0 ex2 dx = /2.
 15.15.69: Show that, if r > 0, then for any positive x, (x) =r x 0 ert t x1 d...
 15.15.70: Show that, for positive x, (x) = 2 0 et2 t 2x1 dt. Hint: Let t = y2...
 15.15.71: Define the beta function by B(x, y) = 1 0 t x1 (1 t)y1 dt. It can b...
 15.15.72: Show that, if x and y are positive integers, then B(x, y) = (x) (y)...
Solutions for Chapter 15: Special Functions and Eigenfunction Expansions
Full solutions for Advanced Engineering Mathematics  7th Edition
ISBN: 9781111427412
Solutions for Chapter 15: Special Functions and Eigenfunction Expansions
Get Full SolutionsAdvanced Engineering Mathematics was written by and is associated to the ISBN: 9781111427412. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 15: Special Functions and Eigenfunction Expansions includes 72 full stepbystep solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 7. Since 72 problems in chapter 15: Special Functions and Eigenfunction Expansions have been answered, more than 26532 students have viewed full stepbystep solutions from this chapter.

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

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.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

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

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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.

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

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.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

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).

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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