FAC1004 Tutorial 3 — Complex Logarithm

Practice problems on complex logarithms, complex trigonometric functions, and geometric loci.

Topics Covered

• Complex logarithm (general and principal)
• Complex trigonometric functions
• Geometric interpretations: circles, perpendicular bisectors, half-lines
• Converting between complex and Cartesian equations

Problem Set

1. Complex Logarithm: Find $\ln(z)$ and $\text{Ln}(z)$ (principal value) for:

   • $z = \frac{1}{2} + \frac{1}{2}i$
   • $z = 2 - \sqrt{3}i$
   • $z = 2\sqrt{3}e^{i\pi/6}$

2. Complex Trigonometric Functions:

   • Find Cartesian form of $\sin i$ and $\cos i$
   • Compute $\cos\left(\frac{\pi}{4} - i\right)$ and $\sin\left(\frac{\pi}{4} - i\right)$ using compound angle formulas

3. Complex Exponential: Find $\text{Re}(z)$, $\text{Im}(z)$, and $\arg(z)$ for $z = 2^{i+3}$

4. Circle Loci: Sketch and find Cartesian equations for:

   • $|z - (4+i)| = 3$
   • $|z - 1 - i| = 5$
   • $|2z + 6 - 4i| = 6$

5. Perpendicular Bisector Loci: Sketch and find Cartesian equations for:

   • $|z - (2+i)| = |z - (1+3i)|$
   • $|z + 2 - i| = |z - 1 + 3i|$

6. Half-line Loci: Sketch and find Cartesian equations for half-lines at various angles

7. Combined Problems: Give both complex and Cartesian equations for:

   • Circle of radius 5 with center at $z = 3 - 2i$
   • Perpendicular bisector of segment connecting $z = -1-2i$ and $z = 3-2i$

8. Circle from Equation: Show that $|z - 2| = 2|z + i|$ represents a circle and find its center and radius

Related

• FAC1004 - Advanced Mathematics II (Computing) — main course page
• Complex Numbers — concept page
• FAC1004 L7-L8 — Complex Equations (Geometric Interpretation) — related lecture

Source File

TUTORIALS_SET_2526/FAC1004 Tutorial 3 25-26.pdf