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MAT334 Complex Variables

Homework 8

Instructions

This homework is based on week 11, including contour integration. Please read the Homework Policies for details on submission policies, collaboration rules, and general instructions.

● Office hours are held throughout the week. During office hours, you may discuss the problems with peers and the Instructor/TA. However, you may not outright share your solutions with your peers.

● Submit your polished solutions on Crowdmark. Do not submit rough work. Organize your work neatly. You may be penalized for work that is disorganized, messy, or illegible. Do not send anything by email. Late submissions are not accepted under any circumstance. Remember you can resubmit anytime before the deadline.

● Show your work and justify your steps on every question, unless otherwise indicated. Partial credit will not be given for answers without justification, unless otherwise indicated.

● Each submission must include a signed copy of the Academic Integrity statement. You can find this statement below. Each member of your group must write out and sign their statement. This will be the first question on Crowdmark. Failure to attach Academic Integrity statements from each group member will result in a grade of zero for the group.

Academic integrity statement

I confirm that:

● I have read and followed the policies described in the Homework Policies.

● I have read and understand the rules for collaboration on problem sets described in the Homework Policies. I have not violated these rules while writing this problem set.

● I understand the consequences of violating the University’s academic integrity policies as outlined in the Code of Behaviour on Academic Matters. I have not violated them while writing this assessment.

By signing this document, I agree that the statements above are true.

1. (10 points) Use the residue theorem to compute 

2. (15 points) Use contour integration to find

As part of your solution, you must prove that the integral converges.

Do not use an antiderivative for this function in your solution. Your solution must use only methods from this course to calculate the integral.

3. (15 points) Let k > 0 and

In this problem we will show that .

(3a) (3 points) Prove that this integral converges. (Hint: you will need to break the real line into 4 pieces.)

(3b) (1 point) Explain why .

(3c) (3 points) Let . Prove that lim.

(3d) (3 points) Show that lim by writing  as a power series centered at 0 and using the definition of the line integral. You may assume that you can integrate term by term.

(3e) (5 points) Conclude that  by integrating over the curve below and taking appropriate limits.