The image contains a list of 13 complex mathematics problems (Đề 1) written in Vietnamese. Here's a summary and translation of the problems for clarification:

Problems Overview:

1. Complex Plane Representation: Represent the complex number $2 - 2\sqrt{3}i$ on the complex plane. Find modulus ($|z|$), argument ($\text{arg}(z)$), and express it in polar form and exponential form.

2. Simplification: Simplify:

   • $\frac{7 - i}{2 + i}$
   • $\text{Re}(e^{\frac{\pi}{3}i})$

3. Complex Number Power Calculation: Calculate $3^{i^{171}} - 2^{123} + i^{10} - i$.

4. Rewrite in Rectangular Form: Rewrite $\frac{(i - \sqrt{3})^{15}}{(\sqrt{3} + i)^7}$ into rectangular form.

5. Roots on the Complex Plane: Find and represent the roots of $\sqrt{1 - i}$ on the complex plane.

6. Logarithms and Cosine:

   • Compute $\ln(-i)$
   • Compute $\cos \frac{\pi}{6}$.

7. Lines on the Complex Plane: Represent and describe:

   • $\text{Im} \frac{z - 1}{z + 1} = 0$
   • $|\text{arg}(z + i)| < \frac{\pi}{2}$
   • $3 \leq |z + 1 - i| \leq 4, -\frac{\pi}{2} \leq \text{arg}(z) \leq \frac{\pi}{2}$.

8. Harmonic Function Proof: Prove $u$ is harmonic and find $f(z)$ given:

   • $u(x, y) = x^2 - y^2 + 2x + 4$
   • $f(i) = 3 + 2i$.

9. Image of Curve: Find the image of the curve $|z| = 4$ under the transformation $w = \frac{2}{z}$.

10. Radius of Convergence: Determine the radius of convergence for the power series: $\sum_{n=0}^{\infty} \frac{1 + i}{2^n} (z - 1)^n.$

11. Singular Points and Residues: Determine the singular points and calculate the residues of $f(z) = \frac{e^z}{1 + z^2}$.

12. Complex Integral: Evaluate the integral: $\int_{|z - 1 - i| = 2} \frac{dz}{(z - 1)^2 (z^2 + 1)}.$

13. Application of Residue Theorem: Use the residue theorem to evaluate: $\int_{-\infty}^{\infty} \frac{\cos x}{x^2 - 6x + 10} \, dx.$

Would you like me to solve a specific problem or provide step-by-step explanations for any of these? Let me know! Here are some related questions for deeper exploration:

1. What are the general steps to find the modulus and argument of a complex number?
2. How do you convert a complex number to polar and exponential form?
3. What is the residue theorem, and how is it applied to real integrals?
4. How can you determine whether a function is harmonic?
5. What are the steps to calculate the radius of convergence of a power series?

Tip: Understanding the geometric interpretation of complex numbers on the complex plane can greatly simplify many problems!