Exercises for Section 12.1.  Fourier Series Representation

Instructions   For Exercises 1-2 and 6-11, find the Fourier series representation.

Exercise 1.     
The graph is shown in Figure 12.4.                     Figure 12.4.

Exercise 2.     
The graph is shown in Figure 12.5.                     Figure 12.5.

Exercise 3.   For Exercises 1 and 2, verify that     by termwise differentiation of the Fourier series representation for  .

Exercise 4.   For Exercise 1, set    and conclude that   .  

Exercise 5.   For Exercise 2, set    and conclude that   .

Exercise 6.       
The graph is shown in Figure 12.6.                     Figure 12.6.

Exercise 7.       
The graph is shown in Figure 12.7.                     Figure 12.7.

Exercise 8.       
The graph is shown in Figure 12.8.                     Figure 12.8.

Exercise 9.       
The graph is shown in Figure 12.9.                     Figure 12.9.

Exercise 10.       
The graph is shown in Figure 12.10.                     Figure 12.10.

Exercise 11.       
The graph is shown in Figure 12.11.                     Figure 12.11.

Exercise 12.   Establish Euler's second formula, Equation (12-3), for the coefficients  .

(12-3)              .  

(c) 2008 John H. Mathews, Russell W. Howell