Question

Q1. Solve the complex equation: sinz 3i Q2. Study the analyticity of the complex function fusing Cauchy-Riemann equations: Izl Q3. Evaluate, by using Cauchy's integral formula, the path integral cosh2 z dz (z-1-i(z-4) where C consists of Iz 3 (counterclockwise) Q4. Using the Residue theorem, integrate counterclockwise around the circle C defined by zl 1.5, the following tan z dz Q5. Find, by using parti ial fraction, the Laurent series of the function with center zo 0 for 1< z<3

Answer 1

Given complex number is sin z = 3i = > e^iz - e^-iz/2i = 3i = > e^iz - e^-iz = (3i) (2i) = > e^iz - 1/e^iz = 6i^2 = > (e^iz)^2 - 1/e^iz = -6 [Therefore i^2 = -1] = > (e^iz)^2 - 1 = -6 e^iz = > (e^iz)^2 + 6e^iz - 1 = 0 which is a quadratic equation where a = 1: b = 6 and c = -1 e^iz = -b plusminus squareroot b^2 - 4ac/2a = > e^iz = -6 plusminus squareroot (6)^2 - 4(1) (-1)/2(1) = > -6 plusminus 2 squareroot 36 + 4 = > e^iz = -6 plusminus squareroot 40/2 = > -6 plusminus 2 squareroot 10/2 = > -3 plusminus squareroot 10 e^iz = squareroot 10 - 3 (or) e^iz = -squareroot 10 -3 = > (squareroot 10, 3) Applying log on both sides. log e^iz = log (squareroot 10 - 3) (or) log e^iz = log (-(squareroot 10 + 3)) iz = log (squareroot 10 - 3) iz = log (-(squareroot 10 + 3)) z = 1/I log (squareroot 10