π = → L L π L π L π L

4.1.2

(1) f is odd ? ak=0

0

(2)

b k

=

2

sinkxdx

=

4

0

k

k : even

k : odd

4.1.3

(a)

x x

= =

- jump : 0 jump : 2

-2

df dx

=

2 (x) - 2 (x +

)

(b)

df dx

=

2

1 2

(1 +

2cosx

+

2cos2x

+

2cos3x

+ L)

-

2

1 2

(1 -

2cosx

+

2cos2x

-

2cos3x

+ L)

= 4 (cosx + cos3x + cos5x + L)

? f(x) = 4 (sinx + sin3x + sin5x + L)

3

5

4.1.4

f( ) 2

=1=

4

(sin(

/2)

+

sin (3 /2) 3

+

sin (5 5

/2)

+ L)

?

=

4(1 -

1 3

+

1 5

-

1 7

+

L)

4.1.5

(b12

+

b

2 2

+

b

2 3

+ L)

=

16 2

1 ( 12

+1 32

+1 52

+ L) =

f(x) 2 dx = + = 2

-

?

2

= 8( 1 12

+1 32

+1 52

+

L)

=

8(1

+

1 9

+

1 25

+ L)

4.1.6

(i) r=1

? u = a 0 + a1cos + b1sin + a 2cos2 + b2sin2 + L

?

u 0 ( )

=

4

( sin 1

+

sin3 3

+

sin5 5

+ L)

? u (r, ) = 4 ( r sin + r3 sin3 + r5 sin5 + L)

1

3

5

(ii) u (0, ) = 0

4.1.16

(i)

u 0 ( ) =

2

( sin 1

+

sin3 3

+

sin5 5

+ L)

+

1 2

? u (r, ) = 2 ( r sin + r3 sin3 + r5 sin5 + L) + 1

1

3

5

2

(ii) u (0, ) = 1/ 2

4.1.23

(i)

c jk

=

1 4 2

f(x, y)e-ijx e -ikx dxdy

- -

(ii)

b jk

=

1 2

f(x, y) sinjx sinky dxdy

- -

4.1.24

(a)

c jk

=

1 4 2

-

(x, y)e-ijx e -ikx dxdy =

-

1 4 2

(b)

c jk

=

1 4 2

-

(x)e -ijx e -ikx dxdy =

-

1 4 2

e -ikydy

=

0 1

-

2

k 0

k

=

0

(c) cos2xcos 2 y = 1 + cos2x 1 + cos2y = 1 + cos2x + cos2y + cos2xcos2y

2

2

44

4

4

? a00 = a20 = a02 = a22 = 1/ 4 ? c00 = 1/4, c20 = c-20 = c02 = c0-2 = 1/8, c22 = c-22 = c2-2 = c-2-2 = 1/16

4.1.26

- u xx - u yy = -

- k 2bklsinkxsinly - k2 + l2

- l2bklsinkxsinly k2 + l2

= bklsinkxsinly = f

4.2.1

1 1 1 1

F4-1

=

1 4

1 1

1

-i -1 i

-1 1 -1

i

,

- 1

- i

F2-1

=

1 2

1 1

1 - 1

(i) f = (1 1 1 1) c = F4-1f = (1 0 0 0)

(ii) f = (1 0 1 0) c = F4-1f = (1/2 0 1/2 0)

(iii) f = (1 -1) c = F4-1f = (0 1)

4.2.2

1 1 1 1

F4 c

=

1 1

i -1

-1 1

-- i1c

1 - i

-1

i

(i) c = (1 1 1 1) F4c = (4 0 0 0) (ii) c = (0 0 1 0) F4c = (1 -1 1 -1) (iii) c = (2 4 6 8) F4c = (20 - 4 - 4i - 4

- 4 + 4i)

4.2.3

1/2 1/2 1/2 1/2

(i)

U = F4 /

4

=

1/2 1/2

i/2 - 1/2

- 1/2 1/2

-

i/2

- 1/2

1/2

- i/2

- 1/2

i/2

(ii) (col2)T (col3) = 1/4 + i/4 - 1/4 - i/4 = 0

1/2 1/2 1/2 1/2

(iii) U -1 =

4F4-1

=

1/2 1/2

- i/2 - 1/2

- 1/2 1/2

i/2

- 1/2

=

U

1/2

i/2

- 1/2

-

i/2

4.2.4

1 3 23 11

(1 2 3) c (3 2 1) = 2 1 32 = 11

3 2 11 14

4.2.5

1 1 1 11 2

(a) f c g = (1

1

1

1)c (1

0

1

0)

=

1 1

1 1

1 1

10 = 2 11 2

1 1 1 10 2

(b) F4 (4cd) = F4 (4(1 0 0 0)(1/2 0 1/2 0)) = F4 (2 0 0 0) = (2 2 2 2)

4.2.10

(a)

(1 1 1 1 0 0 0) (3 3 3 3 0 0 0) = (3 6 9 12 9 6 3) = (3 6 10 2 9 6 3) = (3 7 0 2 9 6 3) = 3702963

(b) When decimals are all equal to 9,

(9 9 L 9 9) (9 9 L 9 9) = (81 81? 2 L 81? (n -1) 81? n 81? (n -1) L 81? 2 81)

4.2.11

(i) eigenvectors and eigenvalues

x1 = (1 1 1 1), x 2 = (1 i -1 - i), x3 = (1 -1 1 -1), x 4 = (1 - i -1 i)

1 = 0, 2 = 2, 3 = 4, 4 = 2

(ii) 1 = 0 ? det C=0 ? singular (iii) f(x) = 2 - eix - e-ix ? f (0) = 0 = 1, f ( /2) = 2 = 2 , f( ) = 4 = 3 , f( 3 /2) = 2 = 4

4.2.13

W3 = W34 = -1/2 + i 3/2, W32 = -1/2 = -1/2 - i 3/2

Same eigenvectors for C and Cf

( ) ( x1 = (1 1 1), x 2 = 1 W3 W32 , x 3 = 1 W32

) W34

(i) for C

1 = 4 + 1 + 1 = 6, 2 = 4 + W3 + W32 = 3, 3 = 4 + W32 + W34 = 3

(ii) for Cf

1 = f 0 + f1 + f 2 , 2 = f 0 + f1W3 + f 2 W32 , 3 = f 0 + f1W32 + f 2 W34

(iii) If all the eigenvalues of Cf are not zeros, then Cf is invertible.

4.2.18

(a) A=LLT

L

L 1/2

1 1/2

= 1/2 1

1 1/2

=

1/2

5/4

1/2

1/2 1

1 1/2 1/2 5/4 1/2

1/2

L

L

1/2

L

(b) l(t)l(t) = (1 + 1 eit )(1 + 1 e-it ) = 1 e-it + 5 + 1 eit = a(t)

2

2

2

42

(c)

L-1 1 = 1 l(t) 1 + 1 eit

= 1 - 1 eit + 1 e 2it - 1 e3it + L 24 8

2

L 0 L

-1/2 1

0L

? 1/4 -1/2 1 0 L

-1/8 1/4 -1/2

1

0

L -1/8 1/4 -1/2 L

 ................
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