246 Chapter 8Complex numbers
Section 8.4
- (i)Express the complex functionf(x) 1 = 13 x
2
1 + 1 (1 1 + 12 i)x 1 + 1 2(i 1 − 1 1)in the form
f(x) 1 = 1 g(x) 1 + 1 ih(x), whereg(x)andh(x)are real.(ii)solveg(x) 1 = 10 ,h(x) 1 = 10 , then
f(x) 1 = 10. (iii)find|f(x)|
2
.
- (i)Express the complex functionf(z) 1 = 1 z
2
1 − 12 z 1 + 13 in the formf(z) 1 = 1 g(x, y) 1 + 1 ih(x, y)
whereg(x, y)andh(x, y)are real functions of the real variables xand y.(ii)Find the
(real) solutions of the pair of equationsg(x, y) 1 = 10 andh(x, y) 1 = 10 , and hence off(z) 1 = 10 ,
(iii)Solvef(z) 1 = 10 directly in terms of zto confirm the results of (ii).
Section 8.5
Express (i)z, (ii)z*, (iii)z
− 1
in exponential formre
iθ:
30.z 1 = 111 − 1 i 31. 32.z 1 = 12 i 33.z 1 = 1 − 3
Express in cartesian formx 1 + 1 iy:
- 3 e
iπ 2435.e
−iπ 23- 2 e
πi 2637.e
iπ 2238.e
3 πi 2239.e
3 πi40.Use Euler’s formulas for cos 1 xand sin 1 xto show that
(i)cos 1 ix 1 = 1 cosh 1 x, (ii)sin 1 ix 1 = 1 i 1 sinh 1 x, (iii)tan 1 ix 1 = 1 i 1 tanh 1 x
41.Expresscos(a 1 + 1 ib)in the formx 1 + 1 iy.
42.Show thatln 1 z 1 = 1 ln 1 |z| 1 + 1 i 1 arg 1 z
43.Use de Moivre’s formula to find the square roots of−i. Locate them on the complex
plane.
44.Find the number obtained fromz 1 = 131 + 12 iby
(i)anticlockwise rotation through 30°,
(ii)clockwise rotation through 30° about the origin of the complex plane.
Section 8.6
Find all the roots and plot them in the complex plane:
48.The wave functions for the quantum mechanical rigid rotor in a plane are
ψ
n(θ) 1 = 1 Ce
inθ, n 1 = 1 0, ±1, ±2,1=
(i)Calculate the ‘normalization constant’ C for which.
(ii)Show that ifm 1 ≠ 1 n.
Section 8.7
Use complex numbers to integrate:
- 50.Z
0
23
∞
exdx
−xZ sin
0
2
∞
exdx
−xcos
Z
0
2
0
πψ θψ θ θ
mnd
() () =
Z
0
2
2
1
πψ θθ
n()d =
1
8
1
5
1
4
zi=+ 3