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Q1:Question 4. Let R > 0 be a real number and let CR denote the semicircular arc of radius R in the upper half of the complex plane centred at the origin from R to -R. -R C R Use lemma 3.2.9 to prove that SCH 1 z² +4 dz 0 as R→ ∞. [5 marks] Question 5. (a) Show that there does not exist a holomorphic function F: C\{0} → C such that F' (z) = 1/z for all non-zero z = C. (b) Deduce that there does not exist a holomorphic function L: C\{0} → C such that exp(L(z)) = z for all non-zero z Є C. [5 marks]See Answer
Q2:6. (9 points) Find the Laurent series of f(z) 2-7 z²+z-6 for 1 <|z-1|<4.See Answer
Q3:z²-2 5. (8 points) Evaluate the contour integral Sc (z+4i)² dz where C is the curve defined by | z | = 5.See Answer
Q4:2. (8 points) Evaluate the contour integral 24 dz +4 where C is the counterclockwise oriented curve | z |= 1.See Answer
Q5:4. (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. z²-2 5. (8 points) Evaluate the contour integral Jc (z+4i)2 dz where C is the curve defined by | z |=5. 2-7 6. (9 points) Find the Laurent series of f(z) = z²+z-6 for 1 < 1 <4.See Answer
Q6:4. (15 points) Evaluate the contour integral (a) fez • f₁z+71/2² (b) f COS Z (z²-4)² dz where C : |z+1|= 5 dz where Cz-2=2See Answer
Q7:*$cz² = 1² 3. (10 points) Evaluate dz where C is the counterclockwise oriented curve |z| = 4.See Answer
Q8:1. (10 points) Prove that the contour integral of a continuous function f is independent of path in a domain D if and only if the contour integral of f around every closed, peicewise smooth curve C in D vanishes, f(z)dz = 0.See Answer
Q9:4. (15 points) Evaluate the contour integral.edz v z = 0 to z = 2. dz where C is the broken line from z = - -3i to z = 0 and then fromSee Answer
Q10:1. (15 points) Let z1 = rei and z2 = Reio be two complex numbers, the prove that (a) log (z121) = logz₁ +log z₂+2kлi (b) log() = log2₁-logz₂+2kri where k is an integer.See Answer
Q11:3. (20 points) Find the Laurent series for the function in the given domain. 1 z² + 2z Z z²-3z+2 (a) (b) for 0 <|z|< 2 for 1 <z-3 <2See Answer
Q12:5. (9 points) Show that the function u(x, y) = x³ − 3xy² + y is harmonic and determine it's harmonic conjugate.See Answer
Q13:2. (8 points) Show that the bilinear map w = maps points on the circle | z — 1 |= 1 to points on the line Re(w) = 12.See Answer
Q14:3. (8 points) Find the limit: lim- 2-1 z-i z+i x y 4. (9 points) Show that the function f(z) = (x + x2 + 12) + (1 - x² + 12 ) i is analytic in any open set not containing (0,0). Also compute f'(z). 5. (9 points) Show that the function u(x,y) = x³-3x²+y is harmonic and determine it's harmonic conjugate.See Answer
Q15:4. (10 points) Find all analytic functions with real part u(x, y)=x²-3xy²+y.See Answer
Q16:3. (10 points) Find domains in which u(x,y) is harmonic. Find all harmonic conjugates. x²+x+y² x² + y² u(x, y) =See Answer
Q17:2. (10 points) If f(z) = u+iv is analytic in a domain D, and if either u(x, y) or v(x,y) is constant in D, prove that f(z) is constant in D.See Answer
Q18: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?See Answer
Q19: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? See Answer
Q20:(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.See Answer

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