Exam (elaborations)
MAT3705 Assignment 4 2024 - DUE 5 September 2024
- Course
- Institution
MAT3705 Assignment 4 2024 - DUE 5 September 2024
[Show more]
Seller
Follow
brighttutor10
Reviews received
Content preview
MAT3705 Assignment 42024 - DUE 5 September
2024
QUESTIONS WITH COMPLETE ANSWERS
[DATE]
[COMPANY NAME]
[Company address]
, MAT3705 Assignment 4 2024 - DUE 5 September 2024
1. Let f(z) = z2 (z−i)4 and g(z) = z2+1 (z−i)4 . Explain why f has a pole of
order 4 at z = i, but g has a pole of order 3 at z = i.
2. Let f(z) = sin z (z − π)2(z + π/2) and let C denote the positively oriented
contour C = {z = 4eiθ ∈ C : 0 ≤ θ ≤ 2π}. (a) Identify the types of isolated
singularities of f and calculate the residues of f at these points. Provide
reasons for your answers. (b) Use Cauchy’s Residue Theorem to calculate
Z C f(z) dz.
3. Let f(z) = (z + 1)2 z(z + 3i)(z + i/3)
(a) What type of isolated singularity is z = −i/3 of the function f? Provide
reasons for your answer.
(b) Calculate Resz=−i/3f(z). 1
(c) Calculate the value of k such that Z 2π 0 1 + cos θ 5 + 3 sin θ dθ = k Z
C f(z) dz, where C is the positively oriented contour C = {z = eit : 0 ≤ t ≤
2π}.
(d) You are told (and do not have to calculate) that Resz=0f(z) = −1 and
Resz=−3if(z) = 12+3i 4 . Calculate the value of Z 2π 0 1 + cos θ 5 + 3 sin θ
dθ.
4. Use Residue Theory to calculate Z ∞ −∞ x2 (x2 + 9)2 dx.
5. Let f(z) = z2 (z + 4)(z2 − 9) . Show that lim R→∞ Re Z CR f(z)ei5z dz
= 0, where CR denotes the positively oriented contour {Reiθ : 0 ≤ θ ≤
π}. Justify all steps.
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller brighttutor10. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $2.50. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
67866 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now