1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

Miscellaneous Exercises 129

a. 2 +4sin( 50 x)−12 cos( 41 x);

c.sin( 4 x+ 2 );

e.cos^3 (x);

b.sin^2 ( 5 x);

d. sin( 3 x)cos( 5 x);

f.cos( 2 x+^13 π).

31.Let the functionf(x)be given in the interval 0<x<1 by the formula

f(x)= 1 −x.

Find (a) a sine series, (b) a cosine series, (c) a sine integral, and (d) a

cosine integral that equals the given function for 0<x<1. In each case,

sketch the function to which the series or integral converges in the inter-

val− 2 <x<2.

32.Verify the Fourier integral

∫∞

0

cos(λq)exp

(

−λ^2 t

)

dλ=

√

π

4 texp

(

−q

2

4 t

)

, t> 0 ,

by transforming the left-hand side according to these steps: (a) Con-

vert to an integral from−∞to∞by using the evenness of the inte-

grand; (b) replace cos(λq)by exp(iλq)(justify this step); (c) complete

the square in the exponent; (d) change the variable of integration; (e) use

the equality

∫∞

−∞

exp

(

−u^2

)

du=

√

π.

33.Approximate the first seven cosine coefficients(ˆa 0 ,ˆa 1 ,...,aˆ 6 )of the
function
f(x)= 1 +^1 x 2 , 0 <x< 1.

34.Use Fourier sine series representations ofu(x)and of the functionf(x)=
x,0<x<a, to solve the boundary value problem

d^2 u

dx^2 −γ

(^2) u=−x, 0 <x<a,
u( 0 )= 0 , u(a)= 0.
35–43.For each of these exercises,
a. find the Fourier cosine series of the function;
b.determine the value to which the series converges at the given values
ofx;