130 Chapter 1 Fourier Series and Integrals
c.sketch the even periodic extension of the given function for at least
two periods.
44–52. For each of these exercises,
a. find the Fourier sine series of the function;
b. determine the value to which the series converges at the given values
ofx;
c.sketch the odd periodic extension of the given function for at least
two periods.- & 44. f(x)=
0 , 0 <x<a
3 ,
x−a
3, a
3<x<^2 a
3, x= 0 ,a
3,a,−a
2,
a
3 ,2 a
3 <x<a.- & 45. f(x)=
1
2
, 0 <x<a
2, x=a
2, 2 a, 0 ,−a,1 , a
2<x<a.- & 46. f(x)=
2 x
a,^0 <x<a
2 , x=^0 ,a
2 ,a,3 a
2 ,
( 3 a− 2 x)
2 a ,a
2 <x<a.- & 47. f(x)=
x, 0 <x<a
2 , x=^0 ,a,−a
2 ,
a
2 ,a
2 <x<a.- & 48. f(x)=
(a−x)
a ,0<x<a, x=^0 ,a,−a
2.- & 49. f(x)=
0 , 0 <x<a
4 ,
1 , a
4<x<^3 a
4, x= 0 ,a
4,a
2,a,−^3 a
4,
0 ,^3 a
4<x<a.- & 50. f(x)=x(a−x), 0 <x<a, x= 0 ,−a,−
a
2.- & 51. f(x)=ekx, 0 <x<a, x= 0 ,
a
2 ,a,−a.