2. Mandatory Question (simple computation) 10 points. Compute the following integrals with the help of the residue theorem. (Do not forget to say where and what type of singularities the corresponding

complex functions have!) (a) I = = ["th (b) I 1 (tan² (t) + 2) (cotan² (t) + 2) 0 = x sin(x) = cos(x) - f x4-1 -∞ Kramers relations! (Do not forget to check the conditions!) dt (Hint: have a look at the Fourier transform on intervals), d dx with f the Cauchy principal value integral. Use the Kronig- 3. Mandatory Question (simple proof) 10 points. Find out how many zeros the following functions have in the given domains. Apply Rouché's Theorem! (a) h(z) = sin(z) + 4[z² − (3 + i)z + 3i] in the disc Ď(0, 2) (Hint: e² ≈ 7.4 and e-² ≈ 0.1); (b) h(z) = e²/2 +2e²+iz – 1 in the rectangle R = {x + iy € C|0 ≤ x ≤ 1, 0 ≤ y ≤ 2} (Hint: 2 √5 2.2 and e√5/2≈ 3.1); 4. Optional Question (advanced computation) 10 points. Compute the real series S = (-1)³ j¹ – 1 j=2 with the help of the residue theorem. Approach as follows: (a) Rewrite the summands in terms of residues of a function f(z) = g(z)/ sin(72). Where and of what type are the singularities? (b) Perform an ML-bound to rewrite the series in terms of a limit of a contour integral. You can use | sin(2)| ≥ | sin(|z|). Write explicitly what S in terms of the contour integral and the residues is. (c) Do not forget to write the result and simplify as much as possible (no decimal approximation, we want to see the exact result)!