evaluate intergral over c e^3z divided by z^2+1 dz where c is modulus z=1/2Give the answer with clear explanation 

To evaluate the contour integral \(\int_C \frac{e^{3z}}{z^2 + 1} \, dz\), where \(C\) is the contour \(|z| = \frac{1}{2}\), we'll use the residue theorem, which is a powerful tool in complex analysis.

### Step 1: Identify the Singularities
The integrand is \(\frac{e^{3z}}{z^2 + 1}\). The function \(z^2 + 1\) has roots at \(z = \pm i\). These are the singularities of the function. Specifically, these singularities are poles of order 1.

- The poles are at \(z = i\) and \(z = -i\).

### Step 2: Check if the Poles are Inside the Contour \(C\)
The contour \(C\) is \(|z| = \frac{1}{2}\), which is a circle of radius \(\frac{1}{2}\) centered at the origin.

- The pole at \(z = i\) has \(|i| = 1\).
- The pole at \(z = -i\) has \(|-i| = 1\).

Both poles \(z = i\) and \(z = -i\) are outside the contour \(|z| = \frac{1}{2}\), so there are **no singularities inside the contour** \(C\).

### Step 3: Apply the Residue Theorem
The residue theorem states that the contour integral of a function around a closed contour \(C\) is \(2\pi i\) times the sum of the residues of the function inside \(C\).

Since there are no singularities inside the contour \(C\), the sum of the residues inside \(C\) is zero. Thus:

\[
\int_C \frac{e^{3z}}{z^2 + 1} \, dz = 0
\]

### Conclusion
The value of the contour integral \(\int_C \frac{e^{3z}}{z^2 + 1} \, dz\), where \(C\) is the contour \(|z| = \frac{1}{2}\), is **0**.

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Would you like further details or have any questions?

Here are 5 related questions for further exploration:
1. How would the answer change if the contour \(C\) was \(|z| = 2\)?
2. What is the residue of \(\frac{e^{3z}}{z^2 + 1}\) at \(z = i\)?
3. How does the residue theorem apply to higher-order poles?
4. Can the residue theorem be used for integrals over contours that are not circles?
5. What is the geometric interpretation of the residue theorem?

**Tip:** When using the residue theorem, always ensure that you correctly identify the poles within the contour to avoid mistakes in calculating the integral.

Learn how to evaluate the contour integral of e^(3z) / (z^2 + 1) over the contour |z| = 1/2 using the residue theorem in complex analysis. Understand the singularities, contour selection, and application of the residue theorem. Suitable for advanced undergraduate students and those interested in complex analysis.

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