Complex Numbers
(e) (-2 + 3i)^3
(g) [ 3 (cos i + i sin i ) r
(i) (2ei1r/4)3- Simplify.
(f) (-1+2i)^4
(h) [ ~ (cos i + i sin i) r
(j) ( 2 e-31ri/2)41 + i + 2i^5 + i^7 1 + 3i^2 + 5i^4 + 7i^6(a) 1+i4+2i6 + iB (b) 2i3 + 4i^5 + 6i^7
- Represent the following geometrically.
(a) (2ei?r/6)3 (b) (ei?r/9)5
(c) (2e2i?r/5)3 (d) (1-i)3
494. If P(z) is a polynomial with real coefficients, prove that P(z) = P(z),
and deduce that if P(a) = 0, then P(a) = 0 also.5. If R( z) is a rational fraction (quotient of two polynomials) with real
coefficients, prove that R(z) = R(z).- Show that (1 + i)^4 m is real and that (1 + i)^4 m+^2 is pure imaginary, m
.being any integer. - Prove that
(1 + i tan B ) m = 1 + i tan mO
1 - i tan() 1 - i tan mO
where m is any integer and () f=. (2k + 1 )7r /2, k = 0, ±1, ±2, ....8. If z = x + iy is a complex integer (i.e., if both x and y are integers)
such that lzl > 1, show that
for Rez > 0
andfor Rez < 0
[R. Spira, Amer. Math. Monthly, 68 (1961), 121]- Use De Moivre's theorem to prove each of the following.
(a) sin30 = 3sin0 - 4sin^3 ()
(b) cos 4() = 8 cos^4 () - 8 cos^2 () + 1
( c) sin 5() = 5 sin () - 20 sin^3 () + 16 sin^5 ()
(d) Slll. 6 () +COS^6 () = 3 cos S^40 + 5(e) zn + z-n = 2cosn0, where z = eiO
- Let Zn = (1 + iVS)n, where n is a positive integer. Prove that
- Show that
hn(znZn-1) = 22 n-^2 y'31-zn+l
l+z+···+zn·= ---
1-z