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.