Answer: In each of 1 through 6 assume that the given function is periodically extended

In each of Problems 1 through 6 assume that the given function is periodically extended outside the original interval. (a) Find the Fourier series for the extended function. (b) Sketch the graph of the function to which the series converges for three periods.

In each of Problems 1 through 6 assume that the given function is periodically extended outside the original interval. (a) Find the Fourier series for the extended function. (b) Sketch the graph of the function to which the series converges for three periods.

In each of Problems 1 through 6 assume that the given function is periodically extended outside the original interval. (a) Find the Fourier series for the extended function. (b) Sketch the graph of the function to which the series converges for three periods.

In each of Problems 1 through 6 assume that the given function is periodically extended outside the original interval. (a) Find the Fourier series for the extended function. (b) Sketch the graph of the function to which the series converges for three periods.

In each of Problems 1 through 6 assume that the given function is periodically extended outside the original interval. (a) Find the Fourier series for the extended function. (b) Sketch the graph of the function to which the series converges for three periods.

In each of Problems 1 through 6 assume that the given function is periodically extended outside the original interval. (a) Find the Fourier series for the extended function. (b) Sketch the graph of the function to which the series converges for three periods.

In each of Problems 7 through 12 assume that the given function is periodically extended outside the original interval. (a) Find the Fourier series for the given function. (b) Let en(x) = f(x) sn(x). Find the least upper bound or the maximum value (if it exists) of |en(x)| for n = 10, 20, and 40. (c) If possible, find the smallest n for which |en(x)| 0.01 for all x.

In each of Problems 7 through 12 assume that the given function is periodically extended outside the original interval. (a) Find the Fourier series for the given function. (b) Let en(x) = f(x) sn(x). Find the least upper bound or the maximum value (if it exists) of |en(x)| for n = 10, 20, and 40. (c) If possible, find the smallest n for which |en(x)| 0.01 for all x.

In each of Problems 7 through 12 assume that the given function is periodically extended outside the original interval. (a) Find the Fourier series for the given function. (b) Let en(x) = f(x) sn(x). Find the least upper bound or the maximum value (if it exists) of |en(x)| for n = 10, 20, and 40. (c) If possible, find the smallest n for which |en(x)| 0.01 for all x.

In each of Problems 7 through 12 assume that the given function is periodically extended outside the original interval. (a) Find the Fourier series for the given function. (b) Let en(x) = f(x) sn(x). Find the least upper bound or the maximum value (if it exists) of |en(x)| for n = 10, 20, and 40. (c) If possible, find the smallest n for which |en(x)| 0.01 for all x.

In each of Problems 7 through 12 assume that the given function is periodically extended outside the original interval. (a) Find the Fourier series for the given function. (b) Let en(x) = f(x) sn(x). Find the least upper bound or the maximum value (if it exists) of |en(x)| for n = 10, 20, and 40. (c) If possible, find the smallest n for which |en(x)| 0.01 for all x.

In each of Problems 7 through 12 assume that the given function is periodically extended outside the original interval. (a) Find the Fourier series for the given function. (b) Let en(x) = f(x) sn(x). Find the least upper bound or the maximum value (if it exists) of |en(x)| for n = 10, 20, and 40. (c) If possible, find the smallest n for which |en(x)| 0.01 for all x.

Periodic ForcingTerms. In this chapter we are concerned mainly with the use of Fourier series to solve boundary value problems for certain partial differential equations. However, Fourier series are also useful in many other situations where periodic phenomena occur. Problems 13 through 16 indicate how they can be employed to solve initial value problems with periodic forcing terms.Find the solution of the initial value problem y + 2 y = sin nt, y(0) = 0, y (0) = 0, where n is a positive integer and 2 = n2. What happens if 2 = n2?

Periodic ForcingTerms. In this chapter we are concerned mainly with the use of Fourier series to solve boundary value problems for certain partial differential equations. However, Fourier series are also useful in many other situations where periodic phenomena occur. Problems 13 through 16 indicate how they can be employed to solve initial value problems with periodic forcing terms.Find the formal solution of the initial value problem y + 2 y =  n=1 bn sin nt, y(0) = 0, y (0) = 0,where > 0 is not equal to a positive integer. How is the solution altered if = m, where m is a positive integer?

Periodic ForcingTerms. In this chapter we are concerned mainly with the use of Fourier series to solve boundary value problems for certain partial differential equations. However, Fourier series are also useful in many other situations where periodic phenomena occur. Problems 13 through 16 indicate how they can be employed to solve initial value problems with periodic forcing terms.Find the formal solution of the initial value problem y + 2 y = f(t), y(0) = 0, y (0) = 0, where f is periodic with period 2 and f(t) = 1, 0 < t < ; 0, t = 0, , 2; 1, < t < 2. See Problem 1.

Periodic ForcingTerms. In this chapter we are concerned mainly with the use of Fourier series to solve boundary value problems for certain partial differential equations. However, Fourier series are also useful in many other situations where periodic phenomena occur. Problems 13 through 16 indicate how they can be employed to solve initial value problems with periodic forcing terms.Find the formal solution of the initial value problem y + 2 y = f(t), y(0) = 1, y (0) = 0, where f is periodic with period 2 and f(t) =  1 t, 0 t < 1; 1 + t, 1 t < 2. See Problem 8.

Assuming that f(x) = a0 2 + n=1 an cos nx L + bn sin nx L , (i) show formally that 1 L L L [f(x)] 2 dx = a2 0 2 + n=1 (a2 n + b2 n). This relation between a function f and its Fourier coefficients is known as Parsevals6 equation. This relation is very important in the theory of Fourier series; see Problem 9 in Section 11.6. Hint: Multiply Eq. (i) by f(x),integrate from L to L, and use the EulerFourier formulas.

This problem indicates a proof of convergence of a Fourier series under conditions more restrictive than those in Theorem 10.3.1. (a) If f and f are piecewise continuous on L x < L, and if f is periodic with period 2L, show that nan and nbn are bounded as n . Hint: Use integration by parts. (b) If f is continuous on L x L and periodic with period 2L, and if f and f are piecewise continuous on L x < L, show that n2an and n2bn are bounded as n . If f is continuous on the closed interval, then it is continuous for all x. Why is this important? Hint: Again, use integration by parts. (c) Using the result of part (b), show that  n=1 |an| and  n=1 |bn| converge. (d) From the result in part (c), show that the Fourier series (4) converges absolutely7 for all x.

Suppose that we wish to calculate values of the function g, where g(x) =  n=1 (2n 1) 1 + (2n 1)2 sin(2n 1)x. (i) It is possible to show that this series converges, albeit rather slowly. However, observe that for large n the terms in the series (i) are approximately equal to [sin(2n 1)x]/(2n 1) and that the latter terms are similar to those in the example in the text, Eq. (6). (a) Show that  n=1 [sin(2n 1)x]/(2n 1) = (/2)  f(x) 1 2  , (ii) where f is the square wave in the example with L = 1. (b) Subtract Eq. (ii) from Eq. (i) and show that g(x) = 2  f(x) 1 2   n=1 sin(2n 1)x (2n 1)[1 + (2n 1)2] . (iii) The series (iii) converges much faster than the series (i) and thus provides a better way to calculate values of g(x).