 Chapter 11.11.2: Fmd the Fourier series of the function f(x), of pedol! p = 2L, and ...
 Chapter 11.11.8: Let f(x) =  I if 0 < x < L f(x) = 1 if 1 < x < 2. f(x) = 0 if x > ...
 Chapter 11.11.3: Are the following functions even. odd. or neither even nor odd?
 Chapter 11.11.9: (Review) Show that 1Ii = i, e ix + eix = 2 cos x, eix  e ;,,, =...
 Chapter 11.11.10: What is a Fourier series? A Fourier sine series? A halfrange expan...
 Chapter 11.11.4: (Calculus review) Review complex numbers.
 Chapter 11.11.5: (Coefficients) Derive the fonnula for en from An and Bn
 Chapter 11.11.1: (Calculus review) Review integration techniques for integrals as th...
 Chapter 11.11.6: Find the trigonometric polynomial F(x) of the form (2) for which th...
 Chapter 11.11.7: Show that the given integral represents the indicated function. Hin...
 Chapter 11.11.2: Fmd the Fourier series of the function f(x), of pedol! p = 2L, and ...
 Chapter 11.11.8: Let f(x) = x if 0 < x < k, f(x) = 0 if x > k. Find Ic(w),
 Chapter 11.11.3: Are the following functions even. odd. or neither even nor odd?
 Chapter 11.11.9: Find the Fourier transform of f(x) (without using Table III in Sec....
 Chapter 11.11.10: Can a discontinuous function have a Fourier series? A Taylor series...
 Chapter 11.11.4: (Even and odd functions) Show that the complex Fourier coefficients...
 Chapter 11.11.5: (Spring constant) What would happen to the amplitudes en in Example...
 Chapter 11.11.1: Theful1damental period is the smallest positive period. Find it for
 Chapter 11.11.6: Find the trigonometric polynomial F(x) of the form (2) for which th...
 Chapter 11.11.7: Show that the given integral represents the indicated function. Hin...
 Chapter 11.11.2: Fmd the Fourier series of the function f(x), of pedol! p = 2L, and ...
 Chapter 11.11.8: Derive formula 3 in Table 1 of Sec. 11.10 by integration.
 Chapter 11.11.3: Are the following functions even. odd. or neither even nor odd?
 Chapter 11.11.9: Find the Fourier transform of f(x) (without using Table III in Sec....
 Chapter 11.11.10: Why did we start with period 27f? How did we proceed to functions o...
 Chapter 11.11.4: (Fourier coefficients) Show that ao = Co, an = Cn + Cn, bn = i(cn ...
 Chapter 11.11.5: (Damping) In Example I change c to 0.02 and discuss how this change...
 Chapter 11.11.1: Theful1damental period is the smallest positive period. Find it for
 Chapter 11.11.6: Find the trigonometric polynomial F(x) of the form (2) for which th...
 Chapter 11.11.7: Show that the given integral represents the indicated function. Hin...
 Chapter 11.11.7: Show that the given integral represents the indicated function. Hin...
 Chapter 11.11.2: Fmd the Fourier series of the function f(x), of pedol! p = 2L, and ...
 Chapter 11.11.8: Find the inverse Fourier cosine transform f(x) from the answer to P...
 Chapter 11.11.3: Are the following functions even. odd. or neither even nor odd?
 Chapter 11.11.9: Find the Fourier transform of f(x) (without using Table III in Sec....
 Chapter 11.11.10: What is the trigonometric system? Its main property by which we obt...
 Chapter 11.11.4: Verify the calculations in Example 1
 Chapter 11.11.5: (Input) What would happen in Example I if we replaced ret) with its...
 Chapter 11.11.1: Show that f = COl1st is periodic with any period but has no fundame...
 Chapter 11.11.6: Find the trigonometric polynomial F(x) of the form (2) for which th...
 Chapter 11.11.7: Show that the given integral represents the indicated function. Hin...
 Chapter 11.11.2: Fmd the Fourier series of the function f(x), of pedol! p = 2L, and ...
 Chapter 11.11.8: Obtain 9';:1(1/(1 + w2 )) from Prob. 3 in Sec. 11.7.
 Chapter 11.11.3: Are the following functions even. odd. or neither even nor odd?
 Chapter 11.11.9: Find the Fourier transform of f(x) (without using Table III in Sec....
 Chapter 11.11.10: What do you know about the convergence of a Fourier . ? senes.
 Chapter 11.11.4: Find further temlS in (9) and graph partial sums with your CAS.
 Chapter 11.11.5: Find a general solution of the ODE y" + w2y = ret) with r(t) as giv...
 Chapter 11.11.1: If f(x) and g(x) have period p, show that hex) = af(x) + bg(x) (a, ...
 Chapter 11.11.6: Find the trigonometric polynomial F(x) of the form (2) for which th...
 Chapter 11.11.7: Show that the given integral represents the indicated function. Hin...
 Chapter 11.11.2: Fmd the Fourier series of the function f(x), of pedol! p = 2L, and ...
 Chapter 11.11.8: Obtain 9';:I(eW ) by integration.
 Chapter 11.11.3: Are the following functions even. odd. or neither even nor odd?
 Chapter 11.11.9: Find the Fourier transform of f(x) (without using Table III in Sec....
 Chapter 11.11.10: What is the Gibbs phenomenon?
 Chapter 11.11.4: Obtain the real series in Example 1 directly from the Euler formula...
 Chapter 11.11.5: Find a general solution of the ODE y" + w2y = ret) with r(t) as giv...
 Chapter 11.11.1: (Change of scale) If f(x) has period 17, show that f(ax), a * O. a...
 Chapter 11.11.6: Find the trigonometric polynomial F(x) of the form (2) for which th...
 Chapter 11.11.7: Represent f(x) as an integral (11).f(x) =o ifO<x<ax>a
 Chapter 11.11.2: Fmd the Fourier series of the function f(x), of pedol! p = 2L, and ...
 Chapter 11.11.8: Find 9'c(1  X2 )1 cos (7TX/2. Hint. Use Prob. 5 in Sec. 11.7.
 Chapter 11.11.3: Are the following functions even. odd. or neither even nor odd?
 Chapter 11.11.9: Find the Fourier transform of f(x) (without using Table III in Sec....
 Chapter 11.11.10: What is approximation by trigonometric polynomials? The minimum squ...
 Chapter 11.11.4: Find the complex Fourier series of the following functions. (Show t...
 Chapter 11.11.5: Find a general solution of the ODE y" + w2y = ret) with r(t) as giv...
 Chapter 11.11.1: Sketch or graph f(x), of period 27T, which for 7T < X < 7T is give...
 Chapter 11.11.6: Find the trigonometric polynomial F(x) of the form (2) for which th...
 Chapter 11.11.7: Represent f(x) as an integral (11). f(x) = 0if x>a
 Chapter 11.11.2: Fmd the Fourier series of the function f(x), of pedol! p = 2L, and ...
 Chapter 11.11.8: Let f(x) = x 2 if 0 < x < I and 0 if x> 1. Find 9'cCf).
 Chapter 11.11.3: Are the following functions even. odd. or neither even nor odd?
 Chapter 11.11.9: Find the Fourier transform of f(x) (without using Table III in Sec....
 Chapter 11.11.10: What is remarkable about the response of a vibrating system to an a...
 Chapter 11.11.4: Find the complex Fourier series of the following functions. (Show t...
 Chapter 11.11.5: Find a general solution of the ODE y" + w2y = ret) with r(t) as giv...
 Chapter 11.11.1: Sketch or graph f(x), of period 27T, which for 7T < X < 7T is give...
 Chapter 11.11.6: Find the trigonometric polynomial F(x) of the form (2) for which th...
 Chapter 11.11.7: Represent f(x) as an integral (11).f(x)if x > 1
 Chapter 11.11.2: Fmd the Fourier series of the function f(x), of pedol! p = 2L, and ...
 Chapter 11.11.8: Does the Fourier cosine transform of XI sin x exist? Of XI cos x?...
 Chapter 11.11.3: Are the following functions even. odd. or neither even nor odd?
 Chapter 11.11.9: Find the Fourier transform of f(x) (without using Table III in Sec....
 Chapter 11.11.10: What do you know about the Fourier integral? Its applications?
 Chapter 11.11.4: Find the complex Fourier series of the following functions. (Show t...
 Chapter 11.11.5: Find a general solution of the ODE y" + w2y = ret) with r(t) as giv...
 Chapter 11.11.1: Sketch or graph f(x), of period 27T, which for 7T < X < 7T is give...
 Chapter 11.11.6: Find the trigonometric polynomial F(x) of the form (2) for which th...
 Chapter 11.11.7: Represent f(x) as an integral (11). f(x) ~ f 1xl2 if o<x< x/2 if <x<2
 Chapter 11.11.2: Fmd the Fourier series of the function f(x), of pedol! p = 2L, and ...
 Chapter 11.11.8: f(x) = 1 (0 < x < (0) has no Fourier cosine or sine transform. Give...
 Chapter 11.11.3: PROJECT. Even and Odd Functions. (a) Are the following expressions ...
 Chapter 11.11.9: Find the Fourier transform of f(x) = xex if x> 0 and o if x < 0 fr...
 Chapter 11.11.10: What is the Fourier sine transform? Give examples.
 Chapter 11.11.4: Find the complex Fourier series of the following functions. (Show t...
 Chapter 11.11.5: Find a general solution of the ODE y" + w2y = ret) with r(t) as giv...
 Chapter 11.11.1: Sketch or graph f(x), of period 27T, which for 7T < X < 7T is give...
 Chapter 11.11.6: CAS EXPERIMENT. Size and Decrease of E*. Compare the size of the mi...
 Chapter 11.11.7: Represent f(x) as an integral (11).f(x) = 0if X>7T
 Chapter 11.11.2: Fmd the Fourier series of the function f(x), of pedol! p = 2L, and ...
 Chapter 11.11.8: Find 9's(e"'X) by integration.
 Chapter 11.11.3: Is the given function even or odd? Find its Fourier series. Sketch ...
 Chapter 11.11.9: Obtain '!F(ex"/2) from formula 9 in Table m.
 Chapter 11.11.10: Find the Fourier series of f(x) as given over one period. Sketch f(...
 Chapter 11.11.4: Find the complex Fourier series of the following functions. (Show t...
 Chapter 11.11.5: Find a general solution of the ODE y" + w2y = ret) with r(t) as giv...
 Chapter 11.11.1: Sketch or graph f(x), of period 27T, which for 7T < X < 7T is give...
 Chapter 11.11.6: (Monotonicity) Show that the minimum square error (6) is a monotone...
 Chapter 11.11.7: Represent f(x) as an integral (11).f(x)if x>a
 Chapter 11.11.2: (Rectifier) Find the Fourier series of the function obtained by pas...
 Chapter 11.11.8: Find the answer to Prob. 11 from (9b).
 Chapter 11.11.3: Is the given function even or odd? Find its Fourier series. Sketch ...
 Chapter 11.11.9: Obtain formula 7 in Table III from formula 8.
 Chapter 11.11.10: Find the Fourier series of f(x) as given over one period. Sketch f(...
 Chapter 11.11.4: Find the complex Fourier series of the following functions. (Show t...
 Chapter 11.11.5: (CAS Program) Write a program for solving the ODE just considered a...
 Chapter 11.11.1: Sketch or graph f(x), of period 27T, which for 7T < X < 7T is give...
 Chapter 11.11.6: Using Parseval"s identity, prove that the series have the indicated...
 Chapter 11.11.7: CAS EXPERIMENT. Approximate Fourier Cosine Integrals. Graph the int...
 Chapter 11.11.2: Show that the familiar identities cos3 x =! cos x + ~ cos 3x and si...
 Chapter 11.11.8: Obtain formula 8 in Table II of Sec. 11.1 I from (8b) and a suitabl...
 Chapter 11.11.3: Is the given function even or odd? Find its Fourier series. Sketch ...
 Chapter 11.11.9: Obtain formula 1 in Table III from formula 2.
 Chapter 11.11.10: Find the Fourier series of f(x) as given over one period. Sketch f(...
 Chapter 11.11.4: Find the complex Fourier series of the following functions. (Show t...
 Chapter 11.11.5: (Sign of coefficients) Some An in Example 1 are positive and some n...
 Chapter 11.11.1: Showing the details of your work, find the Fourier series of the gi...
 Chapter 11.11.6: Using Parseval"s identity, prove that the series have the indicated...
 Chapter 11.11.7: Represent f(x) as an integral (13) f(x) = e if O<x<aif x> a
 Chapter 11.11.2: Obtain the series in Prob. 7 from that in Prob. 8.
 Chapter 11.11.8: Let f(x) = sinx if 0 < x < 7T and 0 if x> 7T. Find 9's(f). Compare ...
 Chapter 11.11.3: Is the given function even or odd? Find its Fourier series. Sketch ...
 Chapter 11.11.9: TEAM PROJECT. Shifting. (a) Show that if f(x) has a Fourier transfo...
 Chapter 11.11.10: Find the Fourier series of f(x) as given over one period. Sketch f(...
 Chapter 11.11.4: PROJECT. Complex Fourier Coefficients. It is very interesting that ...
 Chapter 11.11.5: Find the steadystate oscillation of y" + c/ + Y = r(t) with c > 0 ...
 Chapter 11.11.1: Showing the details of your work, find the Fourier series of the gi...
 Chapter 11.11.6: Using Parseval"s identity, prove that the series have the indicated...
 Chapter 11.11.7: Represent f(x) as an integral (13) f(x) = 0if
 Chapter 11.11.2: Obtain the series in Prob. 6 from that in Prob. 5.
 Chapter 11.11.8: In Table II of Sec. 11.10 obtain formula 2 from formula 4, using r@...
 Chapter 11.11.3: Is the given function even or odd? Find its Fourier series. Sketch ...
 Chapter 11.11.10: Find the Fourier series of f(x) as given over one period. Sketch f(...
 Chapter 11.11.5: Find the steadystate oscillation of y" + c/ + Y = r(t) with c > 0 ...
 Chapter 11.11.1: Showing the details of your work, find the Fourier series of the gi...
 Chapter 11.11.6: Using Parseval"s identity, prove that the series have the indicated...
 Chapter 11.11.7: Represent f(x) as an integral (13)f(x) = 0if x>
 Chapter 11.11.2: Obtain the series in Prob. 3 from that in Prob. 21 of 11.1.
 Chapter 11.11.8: Show that 9'sCx 1I2) = w 1I2 by setting wx = t 2 and using S(oo) ...
 Chapter 11.11.3: Is the given function even or odd? Find its Fourier series. Sketch ...
 Chapter 11.11.10: Find the Fourier series of f(x) as given over one period. Sketch f(...
 Chapter 11.11.5: Find the steadystate oscillation of y" + c/ + Y = r(t) with c > 0 ...
 Chapter 11.11.1: Showing the details of your work, find the Fourier series of the gi...
 Chapter 11.11.6: Using Parseval"s identity, prove that the series have the indicated...
 Chapter 11.11.7: Represent f(x) as an integral (13)f(x) = 0if X> 7T
 Chapter 11.11.2: Using Prob. 3, show that I  ! + ~  k +  . . . = fz7T 2
 Chapter 11.11.8: Obtain 9'sCeax) from (8a) and formula 3 in Table I of Sec. 11.10
 Chapter 11.11.3: Find (a) the Fourier cosine series, (b) the Fourier sine serie~. Sk...
 Chapter 11.11.10: Find the Fourier series of f(x) as given over one period. Sketch f(...
 Chapter 11.11.5: Find the steadystate oscillation of y" + c/ + Y = r(t) with c > 0 ...
 Chapter 11.11.1: Showing the details of your work, find the Fourier series of the gi...
 Chapter 11.11.7: Represent f(x) as an integral (13)f(x)if x> 7T
 Chapter 11.11.2: Show that I +! + ~ + k + ... = ~7T
 Chapter 11.11.8: Show that 9's(x3/2) = 2w1/2 Hint. Set wx = t 2 , integrate by part...
 Chapter 11.11.3: Find (a) the Fourier cosine series, (b) the Fourier sine serie~. Sk...
 Chapter 11.11.10: Find the Fourier series of f(x) as given over one period. Sketch f(...
 Chapter 11.11.5: CAS EXPERIMENT. Maximum Output Term. Graph and discus~ outputs of y...
 Chapter 11.11.1: Showing the details of your work, find the Fourier series of the gi...
 Chapter 11.11.7: Represent f(x) as an integral (13)f(x)if x>a
 Chapter 11.11.2: CAS PROJECT. Fourier Series of 2LPeriodic Functions. (a) Write a p...
 Chapter 11.11.8: (Scale change) Using the notation of (5), show that f(ax) has the F...
 Chapter 11.11.3: Find (a) the Fourier cosine series, (b) the Fourier sine serie~. Sk...
 Chapter 11.11.10: Find the Fourier series of f(x) as given over one period. Sketch f(...
 Chapter 11.11.5: Find the steadystate current I(t) in the RLCcircuit in Fig. 272, ...
 Chapter 11.11.1: Showing the details of your work, find the Fourier series of the gi...
 Chapter 11.11.7: PROJECT. Properties of Fourier Integrals (a) Fourier cosine integra...
 Chapter 11.11.2: CAS EXPERIMENT. Gibbs Phenomenon. The partial sums ,1'n(X) of a Fou...
 Chapter 11.11.8: WRITING PROJECT. Obtaining Fourier Cosine and Sine Transforms. Writ...
 Chapter 11.11.3: Find (a) the Fourier cosine series, (b) the Fourier sine serie~. Sk...
 Chapter 11.11.10: Find the Fourier series of f(x) as given over one period. Sketch f(...
 Chapter 11.11.5: Find the steadystate current I(t) in the RLCcircuit in Fig. 272, ...
 Chapter 11.11.1: Showing the details of your work, find the Fourier series of the gi...
 Chapter 11.11.3: Find (a) the Fourier cosine series, (b) the Fourier sine serie~. Sk...
 Chapter 11.11.10: Using the answers to suitable oddnumbered problems, find the sum of
 Chapter 11.11.1: Showing the details of your work, find the Fourier series of the gi...
 Chapter 11.11.3: Find (a) the Fourier cosine series, (b) the Fourier sine serie~. Sk...
 Chapter 11.11.10: Using the answers to suitable oddnumbered problems, find the sum of
 Chapter 11.11.1: Showing the details of your work, find the Fourier series of the gi...
 Chapter 11.11.3: Find (a) the Fourier cosine series, (b) the Fourier sine serie~. Sk...
 Chapter 11.11.10: Using the answers to suitable oddnumbered problems, find the sum of
 Chapter 11.11.1: Showing the details of your work, find the Fourier series of the gi...
 Chapter 11.11.3: Find (a) the Fourier cosine series, (b) the Fourier sine serie~. Sk...
 Chapter 11.11.10: (Parseval's identity) Obtain the result of Prob. 23 by applying Par...
 Chapter 11.11.1: Showing the details of your work, find the Fourier series of the gi...
 Chapter 11.11.3: Find (a) the Fourier cosine series, (b) the Fourier sine serie~. Sk...
 Chapter 11.11.10: What are the sum of the cosine terms and the sum of the sine terms ...
 Chapter 11.11.1: (Discontinuities) Verify the last statement in Theorem 2 for the di...
 Chapter 11.11.3: Illustrate the formulas in the proof of Theorem I with examples. Pr...
 Chapter 11.11.10: (Halfrange expansion) Find the halfrange sine series of f(x) = 0 ...
 Chapter 11.11.1: CAS EXPERIMENT. Graphing. Write a program for graphing partial sums...
 Chapter 11.11.10: (Halfrange cosine series) Find the halfrange cosine series of f(x...
 Chapter 11.11.1: CAS EXPERIMENT. Order of Fourier Coefficients. The order seems to b...
 Chapter 11.11.1: Without Integration. Show that for a function whose third derivativ...
 Chapter 11.11.10: Compute the minimum square errors for the trigonometric polynomials...
 Chapter 11.11.1: Apply the formulas in Project 28 to the function in Prob. 21 and co...
 Chapter 11.11.10: Compute the minimum square errors for the trigonometric polynomials...
 Chapter 11.11.1: CAS EXPERIMENT. Orthogonality. Integrate and graph the integral of ...
 Chapter 11.11.10: Solve y" + (lly = ret). where Iwl '* o. I. 2 ..... r(t) is 27fperi...
 Chapter 11.11.10: Solve y" + (lly = ret). where Iwl '* o. I. 2 ..... r(t) is 27fperi...
 Chapter 11.11.10: Sketch the given function and represent it as indicated. If you hav...
 Chapter 11.11.10: Sketch the given function and represent it as indicated. If you hav...
 Chapter 11.11.10: f(x) = I + x/2 if 2 < x < o. f(x) = I  x/2 if o < x < 2, ((x) = 0...
 Chapter 11.11.10: f(x) = I  x/2 if 2 < x < O. f(x) = 1  x/2 if o < x < 2, f(x) = ...
 Chapter 11.11.10: f(x) = 4 + x 2 if 2 < x < 0, f(x) = 4  x 2 if o < x < 2, f(x) = ...
 Chapter 11.11.10: f(x) = 4  x 2 if 2 < x < 2. f(x) = 0 otherwise. by a Fourier cosi...
 Chapter 11.11.10: Find the Fourier transform of f(x) = k if a < x < b. f(x) = 0 other...
 Chapter 11.11.10: Find the Fourier cosine transform of f(x) = e2x if x > 0, f(x) = 0...
 Chapter 11.11.10: Find 9' c(e2x) and 9' s(e2x) by formulas involving second derivat...
Solutions for Chapter Chapter 11: Advanced Engineering Mathematics 9th Edition
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter Chapter 11
Get Full SolutionsSince 220 problems in chapter Chapter 11 have been answered, more than 46334 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter Chapter 11 includes 220 full stepbystep solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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

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

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Iterative method.
A sequence of steps intended to approach the desired solution.

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

lAII = 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.

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).

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).

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 •

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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).

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.