From the updated image, we can now see the correct multiple-choice question clearly. Let's re-evaluate the given options based on the image:

Option A

$\oint_{|z|=3} \frac{1}{(z-2)^4} \, dz = 0$

The singularity $z = 2$ is inside the contour $|z|=3$. For a fourth-order pole, the residue theorem applied directly gives:

• For $f(z) = \frac{1}{(z-2)^4}$, the residue at $z = 2$ does not simplify to a simple calculation as it requires the third derivative of the remaining function after the pole. Given that this is a high-order pole, the integral is not zero.

Option B

$\oint_{|z|=2} \frac{1}{z(z-1)} \, dz = 0$

The singularities are $z = 0$ and $z = 1$, both inside the contour $|z| = 2$.

• Residue at $z = 0$: $\frac{1}{z-1} \bigg|_{z=0} = -1$
• Residue at $z = 1$: $\frac{1}{z} \bigg|_{z=1} = 1$

Sum of residues: $-1 + 1 = 0$

Thus, the integral is $2\pi i \times 0 = 0$.

Option C

$\oint_{|z|=2} \frac{1}{(z-3)^4} \, dz = 0$

The singularity $z = 3$ is outside the contour $|z|=2$, so the integral is zero.

Option D

$\oint_{|z|=2} \frac{1}{z^3(z+1)} \, dz = 0$

The singularities are $z = 0$ and $z = -1$, both inside the contour $|z| = 2$.

• Residue at $z = 0$: $\frac{1}{(z+1)} \bigg|_{z=0} = 1$
• Residue at $z = -1$: $\frac{1}{z^3} \bigg|_{z=-1} = -1$

Sum of residues: $1 + (-1) = 0$

Thus, the integral is $2\pi i \times 0 = 0$.

Option E

$\oint_{|z|=3} \frac{z e^z}{(z-1)^2} \, dz = 2e\pi i$

The singularity $z = 1$ is inside the contour $|z| = 3$.

• Second-order pole at $z = 1$: $\text{Residue} = \lim_{z \to 1} \frac{d}{dz} \left[ z e^z \cdot \frac{1}{(z-1)^2} \right] = e$

Thus, the integral should equal $2\pi i \times e$.

Option F

$\oint_{|z|=1} \frac{1}{z(z-2)^3} \, dz = -\frac{3\pi i}{8}$

The singularity $z = 0$ is inside the contour $|z| = 1$.

• Third-order pole at $z = 0$: [ \text{Residue} = \lim_{z \to 0} \frac{d^2}{dz^2} \left[ z^3 \frac{1}{z(z-2)^3} \right] = \lim_{z \to 0} \frac{d^2}{dz^2} \left[ \frac{1}{(z-2)^3} \right] ]

This requires further detailed calculation. Given the teacher's note, it seems this result might be incorrect.

Conclusion

The results provided indicate that options E and F are incorrect. Given that option E was re-evaluated and found consistent, further checking is needed for option F which likely involves intricate residue calculation.

Would you like to go into further detail on any specific integral or have any questions? Here are 5 questions you might consider next:

1. Can you explain the process for finding residues at higher-order poles?
2. How do you compute the contour integrals for higher-order poles?
3. Can you provide a detailed explanation of why certain integrals are zero?
4. How does the residue theorem apply to functions with multiple singularities inside the contour?
5. What are the common pitfalls when applying the residue theorem to complex integrals?

Tip: When dealing with higher-order poles, ensure you carefully perform differentiation and check for correct orders to avoid calculation mistakes.