Let's proceed **step by step** for this matching problem: --- ### **Step 1: Identify the singularities of the integrand** The integrand is: \[ \frac{\cos(3-z)}{\exp(iz^2)z^2(z^2+4)(z+4)^3} \] The singularities (poles) come from the denominator: - \(z^2\) gives a **double pole at \(z=0\)**. - \(z^2+4 = (z+2i)(z-2i)\) gives **simple poles at \(z=2i\) and \(z=-2i\)**. - \((z+4)^3\) gives a **triple pole at \(z=-4\)**. So, the poles are at: \(z=0\) (order 2), \(z=\pm 2i\) (order 1), \(z=-4\) (order 3). **Explanation**: This helps us decide which contour encloses which singularities, crucial for applying the Cauchy Integral Formula (CIF) or its generalizations. --- ### **Step 2: Match each contour to the correct evaluation method** #### a) **C = the unit circle** - Encloses: \(z=0\) (since unit circle is centered at 0, radius 1) - Does **not** enclose \(z=\pm2i\) or \(z=-4\) - \(z=0\) is a **double pole**. By the generalized CIF, for a double pole, you evaluate the **first derivative** at \(z=0\). - **Match:** Option 6: *Evaluate the first derivative of \(\frac{\cos(3-z)}{z^2(z^2+4)(z+4)^3}\) at \(z=0\).* --- #### b) **C = the circle with radius 3 around -5** - Centered at \(-5\), radius 3. - Closest poles: \(z=-4\) (distance 1), \(z=0\) (distance 5), others even further. - Only \(z=-4\) is inside. \(z=-4\) is a **triple pole**. By CIF, for a triple pole, you evaluate the **second derivative** at \(z=-4\). - **Match:** Option 3: *Evaluate the second derivative... at \(z=-4\).* --- #### c) **C = the triangle with vertices 1+i, 100i, and -5** - This is a big triangle in the complex plane. - It will enclose \(z=0\), \(z=2i\), and likely \(z=-4\) but not \(z=-2i\). - Since it encloses multiple poles, **CIF cannot be applied directly**. - **Match:** Option 5: *The integral over this contour cannot immediately be solved with the CIF...* --- #### d) **C = the circle with radius 2 around 1+2i** - Centered at \(1+2i\), radius 2. - The pole at \(z=2i\) (distance from center: \(|(1+2i)-2i| = 1|\)), so it's inside. - Other poles are further away. - \(z=2i\) is a **simple pole**. - **Match:** Option 2: *Evaluate ... at \(z=2i\).* --- #### e) **C = the triangle with vertices -1, 1-i, and 4i** - This triangle encloses \(z=0\) and \(z=2i\), but not others. - Multiple poles, so CIF can't be applied directly. - **Match:** Option 5: *The integral over this contour cannot immediately be solved with the CIF...* --- #### f) **C = the circle with radius 3 around 0** - Centered at 0, radius 3. - Encloses \(z=0\), \(z=2i\), and \(z=-2i\) (\(|2i| = 2 < 3\)), but not \(z=-4\). - Multiple poles, so CIF can't be applied directly. - **Match:** Option 5: *The integral over this contour cannot immediately be solved with the CIF...* --- #### g) **C = the square with vertices -3, -4+i, -5, and -4-i** - This square is centered around \(-4\), so only \(z=-4\) is inside. - \(z=-4\) is a **triple pole**. - **Match:** Option 3: *Evaluate the second derivative... at \(z=-4\).* --- ### **Final Answers:** | Contour | Match | |---------|-------| | 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 | 5 | | C = the circle with radius 3 around 0 | 5 | | C = the square with vertices -3, -4+i, -5, and -4-i | 3 | --- **Summary:** 1. Identify which poles are inside each contour. 2. Use the (generalized) Cauchy Integral Formula for the order of the pole, or note if multiple poles prevent direct application. 3. Match accordingly!

**Final Matching Answers:** - **C = the unit circle** → **6** (evaluate first derivative at \(z=0\)) - **C = circle with radius 3 around \(-5\)** → **3** (evaluate second derivative at \(z=-4\)) - **C = triangle with vertices \(1+i, 100i, -5\)** → **5** (contour encloses multiple poles, CIF not directly applicable) - **C = circle with radius 2 around \(1+2i\)** → **2** (evaluate at \(z=2i\)) - **C = triangle with vertices \(-1, 1-i, 4i\)** → **5** (multiple poles, CIF not directly applicable) - **C = circle with radius 3 around 0** → **5** (multiple poles, CIF not directly applicable) - **C = square with vertices \(-3, -4+i, -5, -4-i\)** → **3** (evaluate second derivative at \(z=-4\))