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MAT420 Final Exam Solutions, Exams of Mathematics

Solutions to the take-home and in-class parts of a mat420 final exam, focusing on integration, residues, complex analysis, and laurent series. Students can use this document to check their answers and understand the concepts covered in the exam.

Typology: Exams

Pre 2010

Uploaded on 08/09/2009

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MAT420 Final: Take Home Part Due December 11, 2007
Prof. Thistleton
Please write all answers in the space provided.
1. Integrate the following.
Z2π
0
1
5cos2(θ)+4
Z2π
0
cos(2θ)
13 12cos(θ)
1
pf3
pf4
pf5

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MAT420 Final: Take Home Part Due December 11, 2007 Prof. Thistleton

Please write all answers in the space provided.

  1. Integrate the following.

∫ (^2) π

0

5 cos^2 (θ) + 4

∫ (^2) π

0

cos(2θ) 13 − 12 cos(θ)

  1. Integrate the following.

C+ 2 (0)

sin(z) z^2 + 1

dz

C+ 1 (0)

z sin^2 (z)

dz

  1. Calculate the residues.

Res

[ sin(z) z^2 + 1

, i

]

Res

[ 1 z sin(z)

]

MAT420 Final: In Class Part December 11, 2007, 3:00-5:00 p.m. Prof. Thistleton

Please write all answers in the space provided.

  1. Integrate the following along the straight line contour from z 0 = 0 + i0 to z 1 = −1 + i ∫

C

z^3 dz

  1. Integrate the following along the circular contour z(t) = ei t^ for 0 ≤ t ≤ π.

C

z

dz

  1. and along the circular contour z(t) = ei t^ for π ≤ t ≤ 2 π.

C

z

dz

  1. Find the image of the circle C 1 ( i 2 under the reciprocal transformation w = (^) z^1.
  2. Let f (z) = (^) (1−z)(2^1 −z). Calculate the Laurent series representation for this function for |z| < 1.