Complex Numbers and Loci in Advanced Mathematics II

Complex Logarithms

General complex logarithm: $\ln(z) = \ln|z| + i(\arg(z) + 2k\pi)$, $k \in \mathbb{Z}$
Principal complex logarithm: $\operatorname{Ln}(z) = \ln|z| + i\operatorname{Arg}(z)$, where $-\pi < \operatorname{Arg}(z) \leq \pi$

Example: For $z = \frac{1}{2} + \frac{1}{2}i$: $$|z| = \sqrt{\frac{1}{2}}, \quad \arg(z) = \frac{\pi}{4} + 2k\pi$$ $$\ln(z) = \ln\frac{1}{\sqrt{2}} + i\left(\frac{\pi}{4} + 2k\pi\right), \quad \operatorname{Ln}(z) = -\frac{1}{2}\ln 2 + i\frac{\pi}{4}$$

Complex Trigonometric Functions

Defined using exponentials: $$\sin z = \frac{e^{iz} - e^{-iz}}{2i}, \quad \cos z = \frac{e^{iz} + e^{-iz}}{2}$$
For purely imaginary arguments: $$\sin(iy) = i\sinh y, \quad \cos(iy) = \cosh y$$

Loci in the Complex Plane

Circle (radius $r$, centre $a+bi$): $$|z - (a+bi)| = r$$ Cartesian: $(x-a)^2 + (y-b)^2 = r^2$
Perpendicular bisector of $(a_1,b_1)$ and $(a_2,b_2)$: $$|z - (a_1+b_1i)| = |z - (a_2+b_2i)|$$ Cartesian: $2(a_2-a_1)x + 2(b_2-b_1)y = a_2^2+b_2^2 - a_1^2-b_1^2$
Half-line from $(a,b)$ at angle $\theta$ to real axis: $$\arg(z - (a+bi)) = \theta$$ Cartesian: $y-b = \tan\theta(x-a)$, $x > a$ (or $x < a$ depending on direction)

Key Results

$|z-2| = 2|z+i|$ represents a circle. Rewriting: $$|z-2|^2 = 4|z+i|^2 \implies (x-2)^2+y^2 = 4(x^2+(y+1)^2)$$ Simplifies to: $3x^2 + 3y^2 + 4x + 8y = 0$, centre $\left(-\frac{2}{3}, -\frac{4}{3}\right)$, radius $\frac{2\sqrt{5}}{3}$