1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

450 Answers to Odd-Numbered Exercises

Figure 1 Graph for Exercise 3, Section 1.7.

is a product of continuous functions and is therefore continuous, except

perhaps where the denominator is 0. Aty=0, cos(^12 y)∼=1, 2 sin(^12 y)∼=y,

soφ(y)∼=|y|^3 /^4 /y=±|y|−^1 /^4 neary=0.

c. Now,

∫π

−πφ^2 (y)dyis finite, so the Fourier coefficients ofφapproach

zero.

Section 1.8

1.aˆ 6 =− 0 .00701, a 6 =− 0 .00569.

3.aˆ 0 = 1. 367 ,

aˆ 1 =− 0. 844 , bˆ 1 =− 0. 043 ,

aˆ 2 = 0. 208 , bˆ 2 =− 0. 115 ,

aˆ 3 = 0. 050 , bˆ 3 =− 0. 050 ,

aˆ 4 = 0. 042 , bˆ 4 = 0. 00 ,

aˆ 5 =− 0. 0064 , bˆ 5 = 0. 043 ,

aˆ 6 = 0. 0167.

Section 1.9

1. Each function has the representations (forx>0)

f(x)=

∫∞

0

A(λ)cos(λx)dλ=

∫∞

0

B(λ)sin(λx)dλ.

a.A(λ)= 2 /π( 1 +λ^2 ),B(λ)= 2 λ/π( 1 +λ^2 );

b.A(λ)=2sin(λ)/π λ,B(λ)= 2 ( 1 −cos(λ))/π λ;

c.A(λ)= 2 ( 1 −cos(λπ ))/λ^2 π,B(λ)= 2 (π λ−sin(λπ ))/π λ^2.

3. a. 1 +^1 x 2 =

∫∞

0

e−λcos(λx)dλ;