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

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 \(z = -\frac{3}{2}\):
\[
\text{Res}\left(f(z), z = -\frac{3}{2}\right) = \frac{-\frac{3}{2} + \frac{3}{2}}{2\left(-\frac{3}{2}\right) + 3} = \frac{0}{-3 + 3} = \frac{0}{0} = \frac{1}{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.

Learn how to evaluate the contour integral ∫(1 / (2z + 3)) dz over the circle |z| = 2 using the residue theorem. Understand the singularity analysis and application of residue theorem in complex analysis. Suitable for advanced undergraduate students.

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