Let's proceed step by step: --- ### **Step 1: Identify the Singularities** First, let's identify the singularities of the integrand: \[ \frac{\cos(3-z)}{\exp(iz^2)z^2(z^2+4)(z+4)^3} \] **Singularities:** - \( z = 0 \) (order 2, since \( z^2 \) in the denominator) - \( z = 2i \) (simple zero of \( z^2 + 4 = (z-2i)(z+2i) \)) - \( z = -2i \) (simple zero of \( z^2 + 4 \)) - \( z = -4 \) (pole of order 3, due to \( (z+4)^3 \)) --- ### **Step 2: Analyze Each Contour** Match each contour to the corresponding option by checking which singularities are inside each contour. 1. **C = the unit circle:** - Centered at 0, radius 1. - Contains: \( z = 0 \) only. - Since \( z=0 \) is a double pole, use the first derivative for Cauchy's formula (generalized version). - **Match:** Option 6. 2. **C = the circle with radius 3 around -5:** - Center: -5, radius 3. - Closest singularity is \( z = -4 \), but check if within radius: - \( |-5 - (-4)| = 1 < 3 \) (yes, \( z = -4 \) is inside). - Other singularities are further away. - Since \( z = -4 \) is a triple pole, need the second derivative. - **Match:** Option 3. 3. **C = the triangle with vertices 1+i, 100i, and -5:** - This large triangle contains \( z = 0, 2i, -2i, -4 \). - All singularities inside. Can't use standard Cauchy formula since multiple poles. - **Match:** Option 5. 4. **C = the circle with radius 2 around 1+2i:** - Center: \( 1+2i \), radius 2. - \( |1+2i - 2i| = |1| = 1 < 2 \), so \( z = 2i \) is inside. - Other singularities: \( z = 0 \) is \( |1+2i| = \sqrt{1^2 + 4} = \sqrt{5} \approx 2.2 \) (outside). - \( z = -2i \): \( |1+2i + 2i| = |1+4i| = \sqrt{1^2 + 16} = \sqrt{17} \approx 4.1 \) (outside). - \( z = -4 \): \( |1+2i+4| = |5+2i| = \sqrt{25+4} = \sqrt{29} \approx 5.4 \) (outside). - Only \( z = 2i \) is inside. It's a simple pole. - **Match:** Option 2. 5. **C = the triangle with vertices -1, 1-i, and 4i:** - This triangle contains \( z = 0 \) and \( z = 2i \) (since 4i is much further up, and -1, 1-i are close to origin). - If both \( z = 0 \) and \( z = 2i \) are inside, but let's check which is more relevant. - Since \( z = 0 \) is a double pole, use first derivative with full denominator. - **Match:** Option 1. 6. **C = the circle with radius 3 around 0:** - Center 0, radius 3. - \( z = 0 \) inside. - \( z = 2i \): \( |2i| = 2 < 3 \) (inside) - \( z = -2i \): \( |-2i| = 2 < 3 \) (inside) - \( z = -4 \): \( |-4| = 4 > 3 \) (outside) - Contains 0, 2i, -2i. - Can't use standard Cauchy formula immediately. - **Match:** Option 5. 7. **C = the square with vertices -3, -4+i, -5, and -4-i:** - This is a square around \( z = -4 \), includes -4. - Only \( z = -4 \) is inside. - Since it's a triple pole, use second derivative. - **Match:** Option 3. --- ### **Final Answers Table** | Contour Description | Option | |----------------------------------------------------------|--------| | C = the unit circle | 6 | | C = the circle with radius 3 around -5 | 3 | | C = the triangle with vertices 1+i, 100i, and -5 | 5 | | C = the circle with radius 2 around 1+2i | 2 | | C = the triangle with vertices -1, 1-i, and 4i | 1 | | C = the circle with radius 3 around 0 | 5 | | C = the square with vertices -3, -4+i, -5, and -4-i | 3 | --- #### **Explanation for Each Step** - **Step 1:** Identified singularities and their order. - **Step 2:** Checked which singularities are inside each contour, and matched to the corresponding evaluation procedure based on pole order and the Cauchy Integral Formula. --- **Final Answer:** | C = the unit circle | 6 | |---------------------|---| | C = the circle with radius 3 around -5 | 3 | | C = the triangle with vertices 1+i, 100i, and -5 | 5 | | C = the circle with radius 2 around 1+2i | 2 | | C = the triangle with vertices -1, 1-i, and 4i | 1 | | C = the circle with radius 3 around 0 | 5 | | C = the square with vertices -3, -4+i, -5, and -4-i | 3 |