Final exam

.pdf
School
Drexel University**We aren't endorsed by this school
Course
ECE 301
Subject
Mathematics
Date
Dec 19, 2024
Pages
3
Uploaded by nnamdinzeduru96
Math 291 - Final - 1/4/2024Name:The following rules apply:This take home exam is due by 12:30 PM, Tuesday Dec 10, in PISB 120. I will be there toreceive your exams from 10:30 to 12:30To get full credit, you must show all work.“Warning”: I have tried solving these problems online, using Chatgpt. It gets the answerswrong. Do not depend on online answers. Even if you get the correct answer, you still needto show your work.However, it might be useful to use programs to simplify simpleparts of the calculation, such as doing complex number arithmetic.Problem 1I want you to evaluate the integralZ011 +x4dxUsing the technique we used in class to evaluateZ011 +x3dxI want you to use the curveγwith three parts: (i)γ1the x-axis from zero toR, (ii)γ2the quartercircleReit,0tπ2, (iii)γ3, the vertical axis from 0 toRi.The closed curve should be orientedcounter-clockwise; be careful with the third part of the integral.(a) (25 points) I am telling you that one root of the denominator isr1=22+i22, and the residueat that point is equal to28i28. Use this to evaluate the integral. You must explain to mewhat happens to the three different parts of the integral.b) (5 points) Calculate the residue atr1.Obviously, you should get the same answer I do!Todo so, you will need to factor the denominator; for this you will need all four roots.
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Math 291Final Exam (take home) - Page 2 of 312/4/2024Problem 2(33 points) Calculate the following integral, using contour integral techniques.Z−∞1(x2+ 1)(x2+ 4)(x2+ 9)dx
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Math 291Final Exam (take home) - Page 3 of 312/4/2024Problem 3(a) (17 points) Consider the complex functionf=zez. Replacezbyx+iy, rewritef=u(x, y) +iv(x, y), and show thatu,vsatisfy the Cauchy-Riemann equations.(b)(16 points) Consider the complex functionf=1ez+ 1. Writef=u(x, y) +iv(x, y) and showthatu,v, satisfy the Cauchy-Riemann equations.
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