To evaluate the contour integral $\int_C \frac{1}{2z + 3} \, dz$, where $C$ is the contour defined by $|z| = 2$, we can proceed as follows:

Step 1: Identify the integrand

The integrand is given by: $f(z) = \frac{1}{2z + 3}$

Step 2: Analyze the singularity

The function $f(z)$ has a singularity where the denominator is zero: $2z + 3 = 0 \implies z = -\frac{3}{2}$ This is a simple pole at $z = -\frac{3}{2}$.

Step 3: Check the location of the singularity relative to the contour

The contour $C$ is the circle $|z| = 2$. The pole $z = -\frac{3}{2}$ lies inside this contour because: $\left| -\frac{3}{2} \right| = \frac{3}{2} < 2$

Step 4: Apply the residue theorem

Since the pole $z = -\frac{3}{2}$ is inside the contour, we apply the residue theorem to evaluate the integral. The residue of $f(z)$ at $z = -\frac{3}{2}$ is given by: $\text{Res}\left(f(z), z = -\frac{3}{2}\right) = \lim_{z \to -\frac{3}{2}} (z + \frac{3}{2}) f(z) = \lim_{z \to -\frac{3}{2}} \frac{z + \frac{3}{2}}{2z + 3}$ Substituting $z = -\frac{3}{2}$:

] ### Step 5: Calculate the integral By the residue theorem, the integral is: \[ \int_C \frac{1}{2z + 3} \, dz = 2\pi i \times \text{Res}\left(f(z), z = -\frac{3}{2}\right) = 2\pi i \times \frac{1}{2} = \pi i$$ So, the value of the integral is: $$\boxed{\pi i}$$ Would you like a more detailed explanation, or do you have any questions? ### Related Questions: 1. What is the residue theorem, and how is it applied in complex analysis? 2. How can you evaluate a contour integral for a function with multiple singularities inside the contour? 3. What are the conditions for a contour integral to be zero using the residue theorem? 4. How does the choice of contour $$C$$ affect the outcome of a contour integral? 5. What is the significance of the residue at a pole in determining the value of a contour integral? ### Tip: In contour integration, always check whether the singularities of the integrand lie inside or outside the contour. This determines whether the residue theorem can be applied and which residues to consider.