Final Exam (3 hours) Applied Complex Varibles, Fall 2013

1 1. Expand f (z) = z(z − 1)(z − 2) in a Laurent series for the domain {z ∈ C | 1 < |z| < 2}. 1 2. Expand f (z) = z2 + 1 in a Taylor series at z = 2i. Compute f (2013)(2i). 3. (a) State Cauchy’s integral formula, (b) Evaluate C ez z2 − 3z + 2 dz, where C is the circle {z ∈ C | |z| = 32}, described in the counterclockwise direction. 4. Prove the identity cos(3z) = 4 cos3(z) − 3 cos(z). 5. Prove that the function f (z) = z z 2 is differentiable at z = 0 but it

Final Exam (3 hours) Applied Complex Varibles, Fall 2013 Expand f (z) = in a Laurent series for the domain {z ∈ C | z(z − 1)(z − 2) 1 < |z| < 2} Expand f (z) = in a Taylor series at z = 2i Compute f (2013)(2i) z +1 (a) State Cauchy’s integral formula, ∫ ez (b) Evaluate dz, where C is the circle {z ∈ C | |z| = 3/2}, − 3z + zdescribed C in the counterclockwise direction Prove the identity cos(3z) = cos3(z) − cos(z) Prove that the function f (z) |= | z z is differentiable at z = but it is not analytic at z = Suppose that z0 is a pole of order m (m ∈ N∗) of function f (z) (a) Prove that there exists an analytic function ϕ defined on a neighborhood of z0 satisfying ϕ(z0) ƒ= such that ϕ(z) f (z) = (z − z0)m (b) Show that lim f (z)∞= z→z0 Let Cρ be the semi-circle { ∈z | |C− z | ≥ = ρ, Im(z) } (ρ > 0), described in the clockwise direction Show that ∫ + ez Σ lim C dz = −πi ρ→0 z−1 z+ + ρ Let Cρ be the circle z = ρ (ρ > 0), described in the { ∈ z | |C| } counterclockwise direction Find the values of the following integerals: (a) ∫ C2 z2013 + dz (b) ∫ C2 (z2 (c) ez dz + 1)(z + 3) ∫ 1 (1 + z + z2)(e + e )dz z z−1 C2 Calculate the following integrals: (a) ∫ +∞ (b) ∫ dx x4 +1x2 + +∞ x2 + dx x4 + +∞ sin(x) dx x (c) ∫ (d ) ∫ +∞ xa dx +4 x2 , where −1 < a < (e ) ∫ +∞ ln(x) dx x2 + (f) ∫ +∞ (g) ∫ +∞ cos(x) dx x2 + sin(x) dx x2 + x + −∞ (h) ∫ +∞ (i) 10 ∫ +∞ x sin(x) dx x2 + 16 1− cos(x) dx x2 Find the poles and residues of the following functions: (a) f1(z) = (z2 + 1)2 (b) f2(z) = cos(z) f2(z) = z sin(z) f3(z) = cos2(z) (c) (d) (e ) 1 f4(z) = z − e −1 z (f) f (z) = tan(z) 11 Find the values of the following integerals: (a) ∫ 2π (b) ∫ + cos(x) dx + sin(x) 2π (c) ∫ 2π dx − cos(x) (d) ∫ 2π (e ) dx (−1 < a < 1) + a cos(x) ∫ π (f) 12 State Rouch´e’s theorem dx + sin(x) dx (a > 0) a2 + sin2(x) ∫ π dx + sin (x) 13 Determine the number of zeros, counting multiplicities, of the following polynomials: (a) z2013 + 2015z2015 + 2011z2011 + in {z ∈ C | |z| < 1} (b) z3 + 4z2 − 2z + in {z ∈ C | |z| < 2} 14 Determine the number of roots, counting multiplicities, of the following equations: (a) z2013 + 6z3 + z + = in the annulus {z ∈ C | < |z| < 2} (b) 0.9e−z − = 2z in the domain {z ∈ C | |z| < 1, Rez < 0} Find the linear fractional transformation that maps the points z1 = 1, z2 15 = 0, z = i onto the points w1 = 0, w2 = i, w3 = 16 Find the image of the upper half plane H = {z ∈ C | Im(z) > 0} under the −z transfor- mation + zw = i 17 Find the image of the semi-infinite trip { D= ∈z | C } Im(z) > under the map w = cos(z) 18 < Re(z) < π, Find a one to one mapping w = f (z) that maps the domain D = {z ∈ C | Im(z) > 0, |z| < 1} \ [i/2, i] onto the upper half plane H = {w ∈ C | Im(w) > 0} ——————–End——————— ... (a) z2013 + 2015z2015 + 2011z2011 + in {z ∈ C | |z| < 1} (b) z3 + 4z2 − 2z + in {z ∈ C | |z| < 2} 14 Determine the number of roots, counting multiplicities, of the following equations: (a) z2013...z2013 + dz (b) ∫ C2 (z2 (c) ez dz + 1)(z + 3) ∫ 1 (1 + z + z2)(e + e )dz z z−1 C2 Calculate the following