 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 7773 students have viewed full stepbystep solutions from this chapter.

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

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Column space C (A) =
space of all combinations of the columns of A.

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].

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

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.

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.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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).

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

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

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

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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

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