https://file.aiquickdraw.com/mathbot/file/1729090479569wo6qux8g.jpg

Let's walk through this complex contour integral problem.

### Problem:
You are asked to find the value of the contour integral  
\[
\oint_C \frac{e^{\frac{\pi i}{3}}}{z^2 - 5z + 2} \, dz
\]  
where \( C \) is the circle \( |z| = 1 \) with positive orientation.

---

### **Step-by-Step Solution:**

#### 1. **Residue Calculation:**
- The integrand is:
  \[
  \frac{e^{\frac{\pi i}{3}}}{z^2 - 5z + 2}
  \]
- The singularities (poles) of this function are the roots of the quadratic \( z^2 - 5z + 2 = 0 \).

  Solving the quadratic equation:
  \[
  z = \frac{5 \pm \sqrt{25 - 8}}{2} = \frac{5 \pm \sqrt{17}}{2}
  \]
  These roots are:
  \[
  z_1 = \frac{5 + \sqrt{17}}{2}, \quad z_2 = \frac{5 - \sqrt{17}}{2}
  \]
  Both roots are **real** and positive. Now, check whether any of these poles lie inside the contour \( |z| = 1 \). Clearly, both roots are larger than 1 (since both are greater than 2).  

  **Conclusion:** There are no poles inside the circle \( |z| = 1 \).

---

#### 2. **Result of the Contour Integral:**
- Since there are no poles inside the contour \( C \), the integral evaluates to **0** by the **Cauchy integral theorem** (which states that if a function is analytic inside and on a simple closed contour, the integral is zero).

---

### **Answer:**
The correct value of the integral is:
\[
0
\]
It seems like neither of the answer options in the screenshot is correct, as both suggest non-zero values. Ensure this was the correct question prompt, and if needed, double-check for typographical errors or clarifications in the original problem. 

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Do you have any further questions, or need any clarification?

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**Related Questions:**
1. What is the Cauchy integral theorem, and how does it apply here?
2. How do you determine whether a root of a quadratic lies inside a contour?
3. What happens if a singularity is exactly on the contour?
4. How do you compute residues of more complex functions?
5. What changes if the contour radius were larger (e.g., \( |z| = 3 \))?

---

**Tip:** When working with contour integrals, always carefully check which singularities are enclosed within the path to avoid calculation mistakes.

The value of ∮_C (e^(πi / 3) / (z^2 - 5z + 2)) dz where C is the circle |z| = 1 with positive orientation.

Solve the complex contour integral ∮_C (e^(πi / 3) / (z^2 - 5z + 2)) dz, where C is the circle |z| = 1 with positive orientation. Learn about contour integrals, quadratic equations, and the Cauchy Integral Theorem.

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