TMA03 - Faculty of Science, Technology, Engineering And Mathematics Complex Analysis - Science and Maths Assignment Help

Assignment Task -                 
 

Task

You can submit this TMA either by post to your tutor or electronically as a PDF file by using the University’s online TMA/EMA service. 

preparing and submitting TMAs, available from the ‘Assessment’ area of the M337 website. 

Your work should be written in a good mathematical style, as demonstrated by the example and exercise solutions in the study units. You should explain your solutions carefully, using appropriate notation and terminology, and write in sentences. As usual, you should simplify algebraic answers where possible. 

In the wording of the questions: 

•  write down or state means ‘write down without justification’ 
•  find, determine, calculate, explain, derive, evaluate or solve means that 

we require you to show all your working in giving an answer 

prove, show or deduce means that you should carefully justify each step of your solution. 

Make sure to reference any significant result from the module materials that you use, and check that all the conditions of the result are satisfied. 

Question 1 (Unit C1) –

(a) Let 

f(z) = z2 + 1 (2z2 + 5z + 2)2. 

(i) Show that f has a pole of order 2 at the point −1/2, and evaluate the residue of f at −1/2. [4] 

(ii) Use the strategy for evaluating real trigonometric integrals and the result from part (a)(i) to deduce that 

Z 2π 0 

(4 cost + 5)2dt = −8π27. [5] cost 

(b) Use a method for summing series of even functions to prove that 

X∞ 16n4 − 1= 4 − π coth π2. [13] 8 n=1 

(c) Use Theorem 5.3 on page 72 of Book C to prove that 

Z ∞ 0 t1/2 t3 + tdt =π√2. [11] 

Question 2 (Unit C2) – 

(a) Sketch the path 

( Γ : γ(t) = −1 + eit (t ∈ [0, 2π]) 1 − e−it (t ∈ [2π, 4π]), 

indicating the directions of increasing values of t. 

(b) Determine the number of zeros of the function 

f(z) = z5 + iz3 − 5z2 + 2 

in each of the following sets. 

(i) S1 = {z : |z| ≤ 1} [5] (ii) S2 = {z : 1 < |z| ≤ 2} [5] (iii) S3 = {z : |z| > 2} [2] 

(c) Let f(z) = (z − π) sin z. Find an integer n such that f is n-to-one near the point π. [3] 

(d) Determine 

max{|exp(1/z3)| : 1 ≤ |z| ≤ 2}, and find all points at which the maximum is attained, giving your answers in Cartesian form. [8] 

(e) Consider the series 

X∞ n=1 z2n 4n + n4. 

(i) Prove that the series is uniformly convergent on {z : |z| ≤ r}, for 0 < r < 2. [4] 

(ii) Prove that the series defines a function that is analytic 

on {z : |z| < 2}. [3] 

This TMA03 - Science and Maths Assignment has been solved by our Science and Maths Experts at My Uni Paper. Our Assignment Writing Experts are efficient to provide a fresh solution to this question. We are serving more than 10000+Students in Australia, UK & US by helping them to score HD in their academics. Our Experts are well trained to follow all marking rubrics & referencing style.

Be it a used or new solution, the quality of the work submitted by our assignment Experts remains unhampered. You may continue to expect the same or even better quality with the used and new assignment solution files respectively. There’s one thing to be noticed that you could choose one between the two and acquire an HD either way. You could choose a new assignment solution file to get yourself an exclusive, plagiarism (with free Turnitin file), expert quality assignment or order an old solution file that was considered worthy of the highest distinction.