1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

72 Chapter 1 Fourier Series and Integrals

2. Show that the functions cos(nπx/a)and sin(nπx/a)satisfy orthogonality
   relations similar to those given in Section 1.


3. Suppose a Fourier series is needed for a function defined in the interval
   0 <x< 2 a. Show how to construct a periodic extension with period 2a,
   and give formulas for the Fourier coefficients that use only integrals from
   0to2a. (Hint: See Exercise 5, Section 1.)


4. Show that the formula
   ex=cosh(x)+sinh(x)
   gives the decomposition of the functionexinto a sum of an even and an
   odd function.


5. Identify each of the following as being even, odd, or neither. Sketch on a
   symmetric interval.
   a.f(x)=x;
   c. f(x)=|cos(x)|;
   e.f(x)=xcos(x);

b. f(x)=|x|;

d. f(x)=arc sin(x);

f. f(x)=x+cos(x+ 1 ).

6. Iff(x)isgivenintheinterval0<x<a, what other ways are there to
   extend it to a function on−a<x<a?


7. Find the Fourier series of these functions.
   a.f(x)=x, − 1 <x<1;
   b.f(x)= 1 , − 2 <x<2;

c. f(x)=

{

x, −^12 <x<^12 ,

1 −x,^12 <x<^32.

8. Is it true that if all the sine coefficients of a functionfdefined on−a
   <x<aare zero, thenfis even?


9. We know that iff(x)is odd on the interval−a<x<a, its Fourier se-
   ries is composed only of sines. What additional symmetry condition onf
   will make the sine coefficients with even indices be zero? Give an exam-
   ple.
   10.Sketch both the even and odd extensions of these functions.
   a.f(x)=1, 0 <x<a;
   c. f(x)=sin(x),0<x<1;

b. f(x)=x,0<x<a;

d. f(x)=sin(x),0<x<π.

11.Find the Fourier sine series and cosine series for the functions given in

Exercise 10. Sketch the even and odd periodic extensions for several peri-

ods.