evaluate integral 4z^2+z+5/(z+1)^2 dz where c : |z|=2

We are asked to evaluate the integral of the function

\[
\int_C \frac{4z^2 + z + 5}{(z+1)^2} \, dz
\]

where \( C \) is the contour \( |z| = 2 \), which is a circle centered at the origin with radius 2.

This is a complex contour integral, and the function \( f(z) = \frac{4z^2 + z + 5}{(z+1)^2} \) has a singularity at \( z = -1 \). Since \( z = -1 \) is within the contour \( C \) (because \( |z| = 2 \) includes \( z = -1 \)), we can apply the **Residue Theorem** to evaluate the integral.

### Step 1: Identifying the singularity

The only singularity inside \( |z| = 2 \) is at \( z = -1 \). We need to find the residue of the function at this point.

### Step 2: Finding the residue at \( z = -1 \)

The function has a pole of order 2 at \( z = -1 \), since the denominator is \( (z+1)^2 \). To find the residue for a second-order pole, we use the formula:

\[
\text{Res}(f, -1) = \lim_{z \to -1} \frac{d}{dz} \left( (z + 1)^2 f(z) \right)
\]

First, let's multiply \( f(z) \) by \( (z + 1)^2 \):

\[
f(z) = \frac{4z^2 + z + 5}{(z+1)^2}, \quad \Rightarrow \quad (z + 1)^2 f(z) = 4z^2 + z + 5
\]

Now, we differentiate \( 4z^2 + z + 5 \) with respect to \( z \):

\[
\frac{d}{dz}(4z^2 + z + 5) = 8z + 1
\]

Next, we evaluate this derivative at \( z = -1 \):

\[
8(-1) + 1 = -8 + 1 = -7
\]

Thus, the residue at \( z = -1 \) is \( -7 \).

### Step 3: Applying the Residue Theorem

By the Residue Theorem, the integral of a function around a closed contour that encloses singularities is \( 2\pi i \) times the sum of the residues of the enclosed singularities. Since there is only one singularity at \( z = -1 \), we have:

\[
\int_C \frac{4z^2 + z + 5}{(z+1)^2} \, dz = 2\pi i \cdot (-7) = -14\pi i
\]

### Final Answer:
\[
\int_C \frac{4z^2 + z + 5}{(z+1)^2} \, dz = -14\pi i
\]

Would you like any further clarification on this, or a more detailed explanation of any step?

Here are five related questions for you to explore:
1. What is the Residue Theorem and how is it used in contour integrals?
2. How do you compute residues for higher-order poles in complex analysis?
3. What are the conditions for applying the Residue Theorem?
4. How do you evaluate integrals involving simple poles versus higher-order poles?
5. Can contour integrals with no singularities inside the contour be non-zero?

**Tip**: When dealing with complex integrals, always check for the type of singularities inside the contour to decide the best evaluation method!

Evaluate the integral 4z^2 + z + 5 / (z+1)^2 dz where c : |z|=2

This solution shows how to evaluate the contour integral ∫_C (4z^2 + z + 5) / (z+1)^2 dz where the contour is |z| = 2. Learn about key concepts in complex analysis like the Residue Theorem and finding residues of higher-order poles.

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