1+2), solve equations in complex numbers and find the limits for functions of complex variable

MTH328 Assignment 1 Value: 10% Overall Task Description You are required to submit a handwritten or typed solution to the following 3 questions which are worth 10 marks each. This assignment covers content from Chapters 1 and 2, and looks at how to express complex numbers in different forms and draw regions described by an expression in complex numbers (question 1+2), solve equations in complex numbers and find the limits for functions of complex variable (question 2), and evaluate derivatives of complex variable functions and find analytical functions satisfying given conditions (question 3). Rationale Assignment one is designed to assess the following learning objectives  Understand the nature of complex numbers, and their representation in the complex plane;  Understand the concept of function as applied to complex numbers, and the ideas of limit, continuity, and differentiation of complex functions. Question 1 (10 marks) 1. Let . Find and . 2. Draw the region described by and and . 3. Draw the region described by and . Marking criteria Question Description Marks 1(1) Correct (1 mark) Incorrect or missing (0 marks) Correct (1 mark) Incorrect or missing (0 marks each) 1(2) Boundary segments of the region Correct (1 mark each segment) Otherwise (0 marks) 1(3) Simplification of Correct (1 mark) Incorrect or missing (0 marks) Boundary segments of the region Correct based on the simplified expression (1 mark each segment) Incorrect or missing (0 marks each) Question 2 (10 Marks) 1. Let . Find , and then express in Cartesian form (to 2 decimal places if necessary). 2. Find all roots of the equation . 3. Check the existence of the following limit . Marking criteria Question Description Marks 2(1) Changing the form of complex number for conveniently evaluating Correct (1 mark) Incorrect or missing (0 marks) Find , , and Correct (1 mark each) Incorrect or missing (0 marks each) 2(2) Changing the form of complex number for conveniently finding roots Correct (1 mark) Incorrect or missing (0 marks) General expression of the roots Correct (1 mark) Incorrect or missing (0 marks) Roots Correct (0.5 marks each) Incorrect or missing (0 marks each) 2(3) Proof or reasons Correct (2 marks) Incorrect or missing (0 marks) Question 3 (10 Marks) 1. Let . a) Find . b) Where does not exist? c) Is it entire? 2. Let . a) Show that is harmonic. b) Find an analytic function such that . c) Find the derivative of . Marking criteria Question Description Marks 3(1) a) Find Correct rules (0.5 marks each) Incorrect or missing (0 marks) Correct result (1 mark) Incorrect or missing (0 marks) b) Find the points where does not exist Correct (1 mark each) Incorrect or missing (0 marks each) c) Is entire? Correct (1 mark) Incorrect or missing (0 marks each) 3(2) a)Prove harmonic Concept and evaluations Correct (0.4 mark each) Incorrect or missing (0 marks each) b)Find analytic function Formulas and evaluations Correct (0.4 mark each) Incorrect or missing (0 marks each) c)Find Correct (1 mark) Incorrect or missing (0 marks each)