FAC1004 Tutorial 3 — Complex Logarithms & Loci
Centre for Foundation Studies in Science
Universiti Malaya
FAC1004 Advanced Mathematics II, 2025/2026
Question 1
Find the general complex logarithm, $\ln(z)$ and principal complex logarithm, $\text{Ln}(z)$ for the following $z$:
(a) $z = \frac{1}{2} + \frac{1}{2}i$
(b) $z = 2 - \sqrt{3}i$
(c) $z = 2\sqrt{3}e^{i\frac{5\pi}{6}}$
Question 2
(a) Find the Cartesian form of $\sin i$ and $\cos i$.
(b) Compute the following complex trigonometric functions by the definition of $\sin z$ and $\cos z$:
(i) $\cos\left(\frac{\pi}{4} - i\right)$
(ii) $\sin\left(\frac{\pi}{4} - i\right)$
Determine (i) and (ii) by applying the compound angle formula of trigonometry, making use of the answer obtained in (a). Show that both methods give the same answer.
Hint: $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$ and $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$
Question 3
Find the $\text{Re}(z)$, $\text{Im}(z)$ and $\arg(z)$ for the complex number $z = 2^{i+3}$.
Hint: $e^{\ln(x)} = x$
Question 4
The complex equation of a circle of radius $r$ with centre at $a + bi$ is given by $|z - (a + bi)| = r$.
Sketch each locus and find the corresponding Cartesian equation.
(a) $|z - (4 + i)| = 3$
(b) $|z - 1 - i| = 5$
(c) $|2z + 6 - 4i| = 6$
Question 5
The complex equation of a perpendicular bisector of the line connecting points $(a_1, b_1)$ and $(a_2, b_2)$ is given by $|z - (a_1 + b_1i)| = |z - (a_2 + b_2i)|$.
Sketch each locus and find the corresponding Cartesian equation.
(a) $|z - (2 + i)| = |z - (1 + 3i)|$
(b) $|z + 2 - i| = |z - 1 + 3i|$
(c) $\left|\frac{z+1+i}{z-2}\right| = 1$
Question 6
The complex equation of a half-line making an angle of $\theta$ with the line parallel to the real axis at the endpoint $(a, b)$ is given by $\arg[z - (a + bi)] = \theta$.
Sketch each locus and find the corresponding Cartesian equation.
(a) $\arg[z - (2 - i)] = \frac{\pi}{4}$
(b) $\arg(z + 2 - i) = \frac{5\pi}{6}$
(c) $\arg(z + 2 + i) = -\frac{\pi}{3}$
Question 7
For each of the following loci, give both its complex and Cartesian equations.
(a) The circle of radius 5 with centre at $z = 3 - 2i$.
(b) The perpendicular bisector of the segment connecting $z = -1 - 2i$ and $z = 3 - 2i$.
(c) The half-line making an angle of $\frac{\pi}{4}$ rad with the line parallel to the real axis at $(-1, -2)$.
Question 8
Show that the locus of point $z$ satisfying the condition $|z - 2| = 2|z + i|$ is a circle. Hence, find its radius and centre.
Key Concepts Covered
- Complex Logarithm — General and principal values
- Complex Trigonometric Functions — sin z and cos z
- Locus — Geometric representations in the Argand diagram
- Circle — Complex equation $|z - z_0| = r$
- Perpendicular Bisector — Equidistant condition
- Half-line — Fixed angle from a point
- Apollonius Circle — Ratio condition $|z - z_1| = k|z - z_2|$