8.5 Exercises 245
The integralIis the real part of this:
(8.55)
The imaginary part is a bonus:
(8.56)
0 Exercises 49, 50
8.8 Exercises
Section 8.2
Express as a single complex number:
1.(2 1 + 13 i) 1 + 1 (4 1 − 15 i) 2.(2 1 + 13 i) 1 + 1 (2 1 − 13 i) 3.(2 1 + 13 i) 1 − 1 (2 1 − 13 i) 4.(5 1 + 13 i)(3 1 − 1 i)
5.(1 1 − 13 i)
2
6.(1 1 + 12 i)
5
7.(1 1 − 13 i)(1 1 + 13 i)
8.Ifz 1 = 131 − 12 i, find (i)zand (ii)zz. (iii)Express the real and imaginary parts of zin terms
of zandz*.
9.Find zsuch thatzz 1 + 1 4(z 1 − 1 z) 1 = 151 + 116 i.
Solve the equations:
10.z
2
1 − 12 z 1 + 141 = 10 11.z
3
1 + 181 = 10
Express as a single complex number:
Section 8.3
(i)Plot as a point in the complex plane, (ii)find the modulus and argument, (iii)Express in
polar formr(cos 1 θ 1 + 1 i 1 sin 1 θ):
- 2 i 17.− 3 18. 11 − 1 i 19. 20.− 61 + 16 i 21.
- 12 i
Givenz
1
andz
2
, express (i)z
1
z
2
, (ii)z
1
2 z
2
, (iii)z
2
2 z
1
as a single complex number for
25.For find (i)z
4
, (ii)z
− 4
.
26.Use de Moivre’s formula to show that
(i)cos 14 θ 1 = 1 cos
4
1 θ 11 − 161 cos
2
1 θ 1 sin
2
1 θ 1 + 1 sin
4
1 θ
(ii)sin 14 θ 1 = 141 sin 1 θ 1 cos 1 θ(cos
2
1 θ 1 − 1 sin
2
1 θ)
27.Use de Moivre’s formula to expandcos 18 xas a polynomial incos 1 x.
zi=+
3
88
cos sin
ππ
zizi
12
5
3
4
3
4
2
3
2
3
=+
cos sin ,=cos +sin
ππ π π
zizi
12
2
22
3
33
=+
,= +
cos sin cos sin
ππ ππ
3 +i −− 212 i
1
5
34
34
−
−
i
i
32
32
−
i
i
1
53 + i
1
1
−
i
i
Zebxdx
ea bxb bx
ab
axaxsin
(sin cos )
=
−
22Zebxdx
ea bxb bx
ab
axaxcos
(cos sin )
=
22