The problem describes a contour integral of a complex function $\frac{z^2 + 9}{z - 4i}$ along a curve $C$, which is parameterized as $z(t) = 3i + 2e^{it}$ for $t \in [-\pi, \pi]$. The integral is given as:

$\int_C \frac{z^2 + 9}{z - 4i} \, dz = -14\pi i$

Steps to Solve:

1. Understand the parameterization:
   The contour $C$ is a circle in the complex plane, centered at $3i$ with radius 2. The parameterization is $z(t) = 3i + 2e^{it}$, which traces the circle once counterclockwise as $t$ varies from $-\pi$ to $\pi$.

2. Singularity inside the contour:
   The function $\frac{z^2 + 9}{z - 4i}$ has a singularity at $z = 4i$ because of the term $\frac{1}{z - 4i}$.

   • To determine whether $4i$ lies inside the contour, compute its distance to the center $3i$:
     $|4i - 3i| = 1$
     Since the radius of the contour is 2, $4i$ lies inside the contour.

3. Apply the residue theorem:
   The integral of a meromorphic function around a closed contour is given by: $\int_C f(z) \, dz = 2\pi i \cdot \text{(Residue of } f(z) \text{ at singularity inside } C).$ Here, $f(z) = \frac{z^2 + 9}{z - 4i}$. The residue at $z = 4i$ is: $\text{Residue} = \lim_{z \to 4i} (z - 4i) \cdot f(z) = \lim_{z \to 4i} \frac{z^2 + 9}{z - 4i} \cdot (z - 4i) = z^2 + 9 \bigg|_{z = 4i}.$

4. Compute the residue:
   Substitute $z = 4i$ into $z^2 + 9$: $z^2 = (4i)^2 = -16, \quad z^2 + 9 = -16 + 9 = -7.$ Thus, the residue is: $-7.$

5. Evaluate the integral:
   Using the residue theorem: $\int_C \frac{z^2 + 9}{z - 4i} \, dz = 2\pi i \cdot (-7) = -14\pi i.$

This matches the given result, confirming the calculation.

Do you want more details or have any questions?

Related Questions:

1. What is the residue theorem, and how is it applied in complex analysis?
2. How do you parameterize other common contours in the complex plane?
3. Why does the residue theorem work only for meromorphic functions?
4. Can a contour integral be evaluated without using the residue theorem? If yes, how?
5. How would the result change if the singularity $4i$ were outside the contour?

Tip:

Always verify whether singularities lie inside or outside the contour before applying the residue theorem. This step is crucial to avoid errors in evaluation!