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)!

Determine whether the function f(2)= 3x+y+L(y-x) is analytic or not. [Page - 76 E2.-1.@ of the book]

9.2 Show that f'(2) does not exist at any pt. where FC2) = 2-2 [Page-70. Ex. 1.6 of the book]

9.3 show that a branch (sec. 33) log z = In 8 + 10 (100) of the logarithmic function can be written log2 = 1/2 In (@a2²77²) +itan (x) 2 in rectangular co-ordinates. Then using the theorem in.. See 23, Show that the given branch is analytic in its domain of definition and that $2 log2 = — there

9.4 Show that Cosliz) = Cos (17) for all z Sin (12) - Sin (iz) if and only if 2. mi (n = 0,1112_) Page-108 Ex-14 of the book)

9.5 Evaluate the integral S(+-i) ³d+ (Page-119 Ex- 2. of the book)

(3) Suppose w(t) = u(t) +iv(t) is complex-valued function of a real variable t and that w'(t) exists. Prove that d/dt zo w(t) = zo w' (t), where zo = xo + i yo is a complex constant.

(4) Let f(z): =1/z (i) Find an anti-derivative F(z) of the function f(z) valid on the region D = {z = re: r>0 and 0 < 0 < 2π}. Sketch the region D.

(1) Suppose w(t) = u(t) +iv(t) is complex-valued function of a real variable t where u(t) and v(t) are real-valued functions on a closed interval bounded piecewise-continuous [a, b]. Prove that

Show that where the c1 is given below in FIGURE 1. The vertices of the triangle in Figure 1 are -1, 1