X y 4. (9 points) Show that the function ƒ(z) = (x+√2 + y² = (x + √₂2² +₂2² ) + ( x − ₁ ² ²+1 1² ) 12+12)i is analytic in any open set not containing (0,0). Also compute f'(z).

Short answers: Q1) Part C (II) C. Let z = 5e^3πi / 8. I. Express 1/z^2 in the form x + yi, where x, ye R. II. Find all rational values of k such that z is purely real. Q3 A. Show that there are no real solutions to the equation et = sec x+i cosec x . B. i. Express ie^it in modulus argument form. ii. Find all integers n such that (sin x+icos x)" = sin nx+cos nx is true for all xe R. C. Let z=(x + yi)³-1li, where x, y, z Z. Using the result in Question 7 (b) (i), or otherwise I. show that 11= y(3x² - y²). II. Hence, find z.

Complex numbers Multiple choice questions Q1 Given z² = 4(cos 3π/2 + i sin3π/2). which of the following is equal to z? Q2 Suppose that z1 and z2 are two complex numbers such that z1z2 = 1. Given that z,1 = re^θ, which of the following is equal to z2? Q3 Suppose a nonzero complex number z1 is multiplied by 1+i / 1-i to produce the complex number z2 Which of the following best describes z2 in relation to z1 ? Q4 Given that z^π =(-1)^3i, which of the following is equal to z? Q5 The Argand diagram below shows the complex number z. Which of the following best represents z²?

Let argo be the branch of arg with argo(z) = (-2, 0]. Let f(z) be the corresponding branch of z. Find the branch cut for f(x³ + 1).

Suppose that f: C→C and g: C→C are continuous. (2a) Prove that is continuous. (2b) Prove, without using any complex limit laws, f g is continuous. Note: citing theorem 2.2.1 from the notes is not sufficient.

Let A = {z = C|0 ≤|z-2i| < 1} U {z E C0 <z + 2i| < 1} U {z € C|Re(z) = 0,-1 ≤ Im(z) ≤ 1}. (4a) Determine whether A is a connected set or not. (4b) Find the smallest subset B C A such that A\B is an open set. (we say B is smaller than C if BCC) (4c) is the set A\B in the previous part connected?

6. (9 points) Find the Laurent series of f(z) 2-7 z²+z-6 for 1 <|z-1|<4.

1 3. (8 points) Evaluate the contour integral ¹ $c (2²-22) ² dz 2. where C' is the counterclockwise oriented curve |z1|=

4. (9 points) Let f be a complex valued function. Prove that if f is analytic in a domain D, then its derivatives f.f",... are analytic in D. 2

z²-2 5. (8 points) Evaluate the contour integral Sc (z+4i)² dz where C is the curve defined by | z | = 5.