EXERCISES 3.3 3.3.1. For the following functions, sketch f(x), the Fourier series of f(x), the Fourier sine series of f(x), and the Fourier cosine series of f(x). (a) f(x) = 1 (b) f(x)=1+x (c) f(x)=x for x<0 and f(x)=1+x for x>0 *(d) f(x)=e^x (e) f(x)=2 for x<0 and f(x)=e^{-x} for x>0 3.3.2. For the following functions, sketch the Fourier sine series of f(x) and determine its Fourier coefficients. (a) f(x)=cos(Pi x/L) Ref: Equation (13) in section H3.3 (b) f(x)=1 for xL/2 (c) f(x)=x for x>L/2 and zero otherwise *(d) f(x)=1 for xL/2 (c) f(x)=x for x>L/2 and zero otherwise 3.3.6. For the following functions, sketch the Fourier cosine series of f (x). Also, roughly sketch the sum of a finite number of nonzero terms (at least the first two) of the Fourier cosine series: (a) f(x) = x [Use formulas (22) and (23) in section H3.3] (b) f(x)=1 for x>L/2 and zero otherwise. [Use carefully formulas (6) and (7) in section H3.3.] (c) f(x)=x for xL/2. [Hint: Add the functions in parts (a) and (b).] 3.3.7. Show that e^x is the sum of an even and an odd function. 3.3.8. (a) Determine formulas for the even extension of any f(x). Compare to the formula for the even part of f(x). (b) Do the same for the odd extension of f(x) and the odd part of f(x). (c) Calculate and sketch the four functions of parts (a) and (b) if f(x)= x for x>0 and f(x)=x^2 for x<0. Graphically add the even and odd parts of f(x). What occurs? Similarly, add the even and odd extensions. What occurs then? 3.3.9. What is the sum of the Fourier sine series of f(x) and the Fourier cosine series of f(x)? [What is the sum of the even and odd extensions of f(x)?] *3.3.10. If f (x) = x^2 for x<0 and f(x)=e^{-x} for x>0, then what are the even and odd parts of f(x)? 3.3.11. Given a sketch of f(x), describe a procedure to sketch the even and odd parts of f(x). 3.3.12. (a) Graphically show that the even terms (n even) of the Fourier sine series of any function on 0 < x < L are odd (antisymmetric) around x = L/2. (b) Consider a function f(x) that is odd around x = L/2. Show that the odd coefficients (n odd) of the Fourier sine series of f(x) on 0 < x < L are zero. *3.3.13. Consider a function f(x) that is even around x = L/2. Show that the even coefficients (n even) of the Fourier sine series of f(x) on 0 < x < L are zero. 3.3.14. (a) Consider a function f(x) that is even around x = L/2. Show that the odd coefficients (n odd) of the Fourier cosine series of f(x) on 0 < x < L are zero. (b) Explain the result of part (a) by considering a Fourier cosine series of f(x) on the interval 0 < x < L/2. 3.3.15. Consider a function f(x) that is odd around x = L/2. Show that the even coefficients (n even) of the Fourier cosine series of f (x) on 0 < x < L are zero. 3.3.16. Fourier series can be defined on other intervals besides -L < x < L. Suppose that g(y) is defined for a < y < b. Represent g(y) using periodic trigonometric functions with period b-a. Determine formulas for the coefficients. [Hint: Use the linear transformation y=(1/2)((a+b) + (b-a)x/L) ] 3.3.17. Consider the integral over x=0 to x=1 of 1/(1+x^2). (a) Evaluate theintegral explicitly. (b) Use the Taylor series of 1/(1 + x^2) (itself a geometric series) to obtain an infinite series for the integral. (c) Equate part (a) to part (b) in order to derive a formula for Pi. 3.3.18. For continuous functions, (a) Under what conditions does f(x) equal its Fourier series for all x, -L < x < L? (b) Under what conditions does f(x) equal its Fourier sine series for all x, 0