1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

94 Chapter 1 Fourier Series and Integrals

3.an=

1

a

∫a

−a

f(x)cos

(nπx

a

)

dx→0asn→∞;

bn=^1 a

∫a

−a

f(x)sin

(

nπx

a

)

dx→0asn→∞.

4.The Fourier series offconverges tofin the sense of the mean.

Properties 2 and 3 are very useful for checking computed values of Fourier
coefficients.

EXERCISES

1.Use properties of Fourier series to evaluate the definite integral

1

π

∫π

−π

(

ln

∣∣

∣∣2cos

(

x

2

)∣∣

∣∣

) 2

dx.

(Hint: See Section 10, Eq. (4), and Section 5, Eq. (5).)

2.Verify Parseval’s equality for these functions:

a.f(x)=x, − 1 <x<1;

b.f(x)=sin(x), −π<x<π.

3.What can be said about the behavior of the Fourier coefficients of the fol-

lowing functions asn→∞?

a.f(x)=|x|^1 /^2 , − 1 <x<1;

b.f(x)=|x|−^1 /^2 , − 1 <x<1.

4.How do we know thatENhas a minimum and not a maximum?

5.If a functionfdefined on the interval−a<x<ahas Fourier coefficients

an= 0 , bn=√^1

n

,

what can you say about

∫a

−a

f^2 (x)dx?

6.Show that, asn→∞, the Fourier sine coefficients of the function

f(x)=^1 x, −π<x<π,

tend to a nonzero constant. (Since this is an odd function, we can take the

cosine coefficients to be zero, although strictly speaking they do not exist.)