Complex Numbers: Logarithms, Trigonometric Functions, Loci

Complex Numbers: Logarithms, Trigonometric Functions, Loci

Logarithm of a Complex Number

• General complex logarithm:
  $$\ln(z) = \ln|z| + i(\arg(z) + 2k\pi), \quad k \in \mathbb{Z}$$
• Principal complex logarithm:
  $$\operatorname{Ln}(z) = \ln|z| + i \operatorname{Arg}(z), \quad -\pi < \operatorname{Arg}(z) \leq \pi$$
• Example: For $z = \frac{1}{2} + \frac{1}{2}i$, $|z| = \frac{\sqrt{2}}{2}$, $\operatorname{Arg}(z) = \frac{\pi}{4}$, so
  $$\operatorname{Ln}(z) = \ln\frac{\sqrt{2}}{2} + i\frac{\pi}{4}$$

Trigonometric Functions of Complex Numbers

• Definitions:
  $$\sin z = \frac{e^{iz} - e^{-iz}}{2i}, \quad \cos z = \frac{e^{iz} + e^{-iz}}{2}$$
• Key values:
  $\sin i = i \sinh 1$, $\cos i = \cosh 1$
• Compound angle formulas (valid for complex numbers):
  $$\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$$
  $$\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$$

Loci in the Complex Plane

• Circle: Centre at $a+bi$, radius $r$
  $$|z - (a+bi)| = r$$
• Perpendicular bisector: Points equidistant from $z_1 = a_1+b_1i$ and $z_2 = a_2+b_2i$
  $$|z - z_1| = |z - z_2|$$
• Half-line: From endpoint $a+bi$ at angle $\theta$ to positive real axis
  $$\arg(z - (a+bi)) = \theta$$

Cartesian Equations from Loci

• Circle $|z - (a+bi)| = r$: $(x-a)^2 + (y-b)^2 = r^2$
• Perpendicular bisector: Solve $|z - z_1|^2 = |z - z_2|^2$ to get linear equation
• Half-line: $y - b = \tan\theta (x-a)$, with domain restriction $x > a$ if $\theta$ is acute, etc.

Transformation to Circle

• Condition $|z - 2| = 2|z + i|$ can be squared and simplified to a circle equation.
  Example result: centre at $\left(-\frac{2}{3}, \frac{4}{3}\right)$, radius $\frac{2\sqrt{5}}{3}$