Complex Analysis Answers

The transformation T : z → w in the complex plane is defined by w =
az + b z + c
, where a, b, c ∈ R.
Given that w = 3i when z = −3i, and w = 10 − 4i when z = 1 + 4i, find the values of a, b and c.
(a) Show that the points for which z is transformed to z lie on a circle and give the centre and radius of this circle.
(b) Show that the line through the point z = 4 and perpendicular to the real axis is invariant under T.

Let x, y, u, v be real numbers and z, w be complex numbers such that z = x + iy, w = u+iv. In each case, describe the locus of the point (u, v) in the Argand diagram.
(a) w = z + 4 and (i) |z| = 3, (ii) arg z = π/3, (iii) |z + 4| = 5, (iv) y2 = 4x

In an Argand diagram, the point P represents the complex number z, where z = x+iy. Given that z+2 = λi(z+8), where λ is a real parameter, find the Cartesian equation of the locus of P as λ varies. If also z = µ(4 + 3i), where µ is real, prove that there is only one possible position for P.

Find two Laurent series in powers of z gor the function f defined by f(z)=1/z^2(1-z) and specify the regions in which the series converges to f(z).

Express −1+i in polar form. Hence show that (−1+i)16 is real and that 1/(−1+i)6 is purely imaginary, giving the value for each.

Use De Moivre’s theorem to simplify the following (a) (cosπ/5+isinπ/5)10, (b) (cosπ/9+isinπ/9)−3, (c) {cos(−π/6) + isin(−π/6)}−4

. If z1 = 1 + i
√
3 and z2 = 2i, find |z1| and arg z1, |z2| and arg z2. Using an Argand
diagram, deduce that arg(z1 + z2) = 5π/12. Hence show that tan(5π/12) = 2 + √
3

Find the modulus and the principal argument of each of the given complex numbers.
(a) 3−4i, (b) −2+i, (c) 1√ , (d) 7−i , 1+i 3 −4−3i
(e) 5(cos π/3 + i sin π/3), (f) cos 2π/3 − sin 2π

Find the modulus and the principal arguement of each of the given complex numbers
a. 3-4i b. -2+i c. 1/1+i(2)½ d. 7-i/-3-3i

Express sin 5 threat and cos5theta in the forms of sin theta