Page 22 - Calculus with Complex Numbers

Exam ples
Hxpress the following complex numbers in the form .x + iy.

(i) (1 + 3ï) + (5 + 7ï), (ii) (1 + 3ï) - (5 + 7ï), (iii) (1 + 3ï)(5 + 7ï),
1 + 3i
(iv) 5 + 7/ , (v) 3 + 4/ , (vi) logtl + ï).

Hint For (vi) use the polar form.
Find 1 + i . Hence show tan zr/8 = .vC - 1.
Hxpand (cos 0 + i sin p)3 to obtain formulae for cos 30, sin 30 in terms of
cos 0 sin 0. Use these formulae to show

cos 3p = 4 cosSp - 3 cos p
,
sin 3p = 3 sin p - 4 sinSp .

Use Question 3 to show that cos 30O = VXz, sin 30O = 1/2.
Hxpand (eio + e-i0)3 to show

4
cos3p = 1(3 cos p + cos 3p),
sin3p = 1(3 sin p - sin 3p).
4
x/2 x/2
Use Question 5 to evaluate cosS 0 d0 sinS 0 d0.
0 0
%
Hvaluate the integral elx cos 4 .x dx by taking the real part of
0

Now do itby integrating by parts twice and compare the efliciency of the two
methods.