3\oint \frac{1}{z-3}dz  C:{z\epsilon \mathbb{C}/\left | z \right |=2;  0\leq Arg(z)\leq 2\pi }  Resolver

To solve the given complex integral

\[3\oint \frac{1}{z-3} \, dz \]

where \(C\) is the contour \(|z|=2\) with \(0 \leq \text{Arg}(z) \leq 2\pi\), we can use the residue theorem.

### Step-by-Step Solution:

1. **Identify the singularity:**
   The integrand \(\frac{1}{z-3}\) has a singularity at \(z = 3\).

2. **Determine if the singularity is inside the contour:**
   The contour \(C\) is given by \(|z| = 2\). The point \(z = 3\) is outside this contour because \(|3| = 3\), which is greater than 2.

3. **Apply the residue theorem:**
   The residue theorem states that for a function \(f(z)\) analytic inside and on a simple closed contour \(C\), except for isolated singularities inside \(C\),

   \[\oint_C f(z) \, dz = 2\pi i \sum \text{Residues of } f(z) \text{ inside } C.\]

   Since the singularity at \(z = 3\) is outside the contour, there are no residues of \(\frac{1}{z-3}\) inside the contour.

4. **Evaluate the integral:**
   Given that there are no singularities inside the contour \(C\),

   \[ \oint_C \frac{1}{z-3} \, dz = 0. \]

   Therefore,

   \[ 3 \oint \frac{1}{z-3} \, dz = 3 \cdot 0 = 0. \]

So, the value of the integral is \(\boxed{0}\).

Do you want more details or have any questions?

Here are 5 questions you might ask next:
1. Can you explain the residue theorem in more detail?
2. How do you determine if a point is inside a contour in the complex plane?
3. What is a singularity in a complex function?
4. Can you provide examples of other complex integrals?
5. How do you handle integrals with singularities on the contour itself?

**Tip:** When solving complex integrals, always check if the singularities of the integrand lie inside the given contour before applying the residue theorem.

Learn how to solve the complex integral \( \oint \frac{1}{z-3} \, dz \) using the residue theorem. Detailed step-by-step solution provided with explanation of contour integration and singularity analysis. Suitable for advanced undergraduate students and those interested in complex analysis.

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