evaluate integral over C 1 by 2 Z + 3 dz  where C is modulus of z equal to 2

To evaluate the integral

\[
\oint_C \frac{1}{2z+3} \, dz
\]

where \( C \) is the circle \(|z| = 2\), we can use the method of complex analysis, specifically the **Cauchy Integral Formula**. First, let's rewrite the integrand in a more convenient form.

### 1. Rewrite the Integrand

The integrand is:

\[
\frac{1}{2z+3}
\]

Let's express this as:

\[
\frac{1}{2z+3} = \frac{1}{2(z + \frac{3}{2})}
\]

\[
= \frac{1}{2} \cdot \frac{1}{z + \frac{3}{2}}
\]

### 2. Apply the Cauchy Integral Formula

The Cauchy Integral Formula states that for a function \( f(z) \) analytic inside and on a simple closed contour \( C \), and for any point \( a \) inside \( C \):

\[
f(a) = \frac{1}{2\pi i} \oint_C \frac{f(z)}{z-a} \, dz
\]

In this problem:

- \( f(z) = \frac{1}{2} \)
- \( a = -\frac{3}{2} \) (since \( z + \frac{3}{2} = 0 \) gives \( z = -\frac{3}{2} \))

The circle \( C: |z| = 2 \) includes the point \( z = -\frac{3}{2} \), so we can directly apply the Cauchy Integral Formula.

### 3. Evaluation of the Integral

Using the Cauchy Integral Formula:

\[
\oint_C \frac{\frac{1}{2}}{z + \frac{3}{2}} \, dz = 2\pi i \cdot \frac{1}{2} \cdot \frac{1}{1}
\]

\[
= \pi i
\]

Thus, the value of the integral is:

\[
\oint_C \frac{1}{2z+3} \, dz = \pi i
\]

### Conclusion

The integral of \(\frac{1}{2z+3}\) over the contour \(C\) where \(|z| = 2\) is:

\[
\pi i
\]

Would you like more details or have any questions about this solution? 

Here are some additional questions related to this topic:

1. How would the integral change if \( C \) was the circle \(|z| = 1\)?
2. What is the significance of the Cauchy Integral Formula in complex analysis?
3. How would you compute the integral if the contour \( C \) did not enclose the singularity?
4. How does the Cauchy-Goursat theorem relate to the evaluation of this integral?
5. Can the same method be used if the integrand had a higher-order pole?

**Tip:** When applying the Cauchy Integral Formula, always check if the singularity lies inside the contour of integration.

Learn how to evaluate the integral ∮_{|z|=2} \frac{1}{2z+3} dz using the Cauchy Integral Formula. Understand the significance of contour integration in complex analysis. Suitable for advanced undergraduate students studying complex analysis.

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