Most of these formulas can be rederived in just a few steps, but it's helpful to know they're out there. Some can save you time, reduce algebra, and highlight physical concepts. Let z be a complex …

z + w becomes the vector sum of P (x; y) and Q(s; t); and jz wj is exactly the distance between P (x; y) and Q(s; t): Henceforth, then, we refer to the xy-plane as the complex plane, and the x-axis as the …

Lemma Let f be a one-to-one holomorphic function f : D ! D(0; 1), f(p0) = 0; f0(p0) > 0, f not onto, then there exists a holomorphic g : D ! D(0; 1), g one-to-one, g(p0) = 0, g0(p0) > f0(p0). Proof: f(z) w As f …

The value of the square root function on the point with coordinates z, w on the parabola w2 = z is w. Notice that if z is given, then √ there are usually two corresponding values of w. Thus if we want to …

Prove that z/w is also a unit complex number. The neat thing about unit complex numbers is that you can multiply and divide them and you always get another unit complex number. If you plot all the unit …

Something of the form x + iy, where x and y are real numbers.

Write in the \algebraic" form (a + ib) the following complex numbers 1; i + i5 = z w = (3 + 3i)8: