**Complex Numbers**

__Things you should know:__

e^{it} = cost + i sint

z = r(cost + i sint) = r.e^{it}, where t=argz and r=|z|.

z^{n} = r^{n}(cost + i sint)^{n} = r^{n}(cos(nt) + i sin(nt)). This is De Moivre's theorem.

z^{n} + z^{-n} = 2 cos(nt)

z^{n} - z^{-n} = 2i sin(nt)

cos(iz) = cosh(z)

sin(iz) = i sinh(z)

|z/w| = |z|/|w|

|zw| = |z| |w|

arg(z/w) = argz - argw

arg(zw) = argz + argw

__Roots of Unity__

1 = cost + i sint => t=0, 2pi, 4pi, ...

1^(1/n) = cos(t/n) + i sin(t/n), where t=0, 2pi, 4pi, ...

So, the nth roots of unity can be written as:

1, w, w^{2}, w^{3}, ..., w^{n-1}, where w = cos(2pi/n) + i sin(2pi/n).

Example:

The fourth roots of unity (n=4; 1^(1/4)) are given by:

1

cos(2pi/4) + i sin(2pi/4) = i

cos(4pi/4) + i sin(4pi/4) = -1

cos(6pi/4) + i sin(6pi/4) = -i

__Roots of Complex Numbers__

The nth roots of any complex number z are given by:

z^{1/n} = r^{1/n} {cos[(t + 2kpi)/n] + i sin[(t + 2kpi)/n]}

Unless a range of values is specified for t, k=0, 1, 2, ..., n-1. If t was allowed to be negative, then you can let k start from negative values, e.g.

You want to find the 3rd roots of z, and you found that z^(1/3) = 4.e^i[(pi/3) + 0.5kpi]. You're given that -pi < t <= pi, so:

k = -2, -1, 0, 1, 2. Because if k=-3, then t = (pi/3) + 0.5kpi = -7pi/6 < -pi, which is out of the range of allowed values, but k=-2 gives t = -2pi/3 > -pi, which is allowed.

__Loci__

Basic loci:

|z - (a+bi)| = k is a circle center (a, b) with radius k.

arg[z - (a+bi)] = t is a half-line starting from (a, b) and making an angle t with the horizontal there.

|z - (a+bi)| = |z - (c+di)| is the perpendicular bisector of the line joining (a, b) and (c, d).

arg[{z - (a+bi)}/{z - (c+di)] = t is an arc of the circle passing through points (a, b) and (c, d) such that the angle suspended on the circumference of the circle by the line connecting these points is equal to t. For negative t, simply draw the locus as if it was for positive t, then the locus you want is the reflection about the line connecting (a, b) and (c, d).

If you're stuck you can simlpy substitute z=x+iy and find an equation of the locus using algebra, but be warned that sometimes a lot of work is required.

__Transformations__

The genearl method is to use the information you're given in the question, e.g.

If you're given a relation: w = f(z), and then you're told that |z|=2, then you should write z in terms of w then take the modulus of both sides of the equation.

Another common way to approach a question is to write w and z in complex number form and equate real and imaginary parts, then use the information given. Remember that you can write a complex number in basic a+bi form, exponential form or modulus-argument form. E.g.

w = f(z) is a transformation that maps z onto w. Show that if:

(a) w lies on the real axis, the point representing z maps a circle;

(b) w lies on the imaginary axis, the point representing z maps a straight line.

The way I'd approach the question is by writing w=u+iv and z=x+iy, and then I'd simplify the equation into a form similar to:

w = u + iv = g(x, y) + i h(x, y)

Then I'd equate real and imaginary parts, i.e. u=g(x, y) and v=h(x, y).

(a) Let's use the information given in the question. If w lies on the real axis, then Im(w)=0, i.e. v=h(x, y)=0. Simplifying this should give us an equation for y in terms of x, and if our working was correct, the equation would be that of a straight line.

(b) Similarly, Re(w)=0, so u=g(x, y)=0, etc.