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MAT420 Exam 1 Solutions - Complex Analysis, Exams of Mathematics

State University of New York Polytechnic - Utica-RomeMathematics

Solutions to exam 1 for the mat420 complex analysis course, due on october 16, 2007. Topics covered include complex multiplication, modulus and argument, cube roots, polar form, transformations, and limits.

Typology: Exams

Pre 2010

Uploaded on 08/09/2009

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koofers-user-9ih 🇺🇸

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MAT420 Exam 1 Due October 16, 2007
Prof. Thistleton
1. Calculate the following and express your result in the form x+iy
(a) (5 + i2)(2 −i4)
3 + i6
(b) Re(1
4+i2)−Im(1
2+i4)
2. Do any of the following points lie inside D2(2 + i3)? Explain.
(a) 1 + i2
(b) 4 −i
3. Let z1= 2 + i3, z2= 3 + i4 and z3=−1−i2. Show that |P3
k=1 zk| ≤ P3
k=1 |zk|
pf3
pf4
pf5

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MAT420Prof. Thistleton1. Calculate the following and express your result in the form Exam 1 (^) x + iy Due October 16, 2007 (a) (5 +^ i3 +2)(2 i 6 −^ i4) (b) Re( (^) 4+^1 i 2 ) − Im( (^) 2+^1 i 4 )

  1. Do any of the following points lie inside(a) 1 + i 2 D 2 (2 + i3)? Explain. (b) 4 − i
  2. Let z 1 = 2 + i3, z 2 = 3 + i4 and z 3 = − 1 − i2. Show that | ∑^3 k=1 zk| ≤ ∑^3 k=1 |zk|
  1. Calculate(a) 1 + i Arg(z) for (b) 2 − i 3 (c) (^) 1+^1 i
  2. Express the following as (a) 5eiπ/ 4 x + iy (b) − 4 e−i^5 π/^6
  3. Calculate Arg(ei^5 π/^6 eiπ/^2 ).
  1. What is the image of the lineState your result analytically and provide a quick sketch. y = x under the transformation f (z) = (1 − i)z + (2 − i3)?
  2. What is the image ofanalytically and provide a quick sketch. D 2 (2 − i) under the mapping f (z) = (3 − i4)z − 2? State your result
  3. Find the image of the set f (z) = z (^2). State your result analytically and provide a quick sketch. {z = reiθ^ : 1 < r < 2 , π/ 4 < θ < pi/ 2 } under the transformation
  4. Calculate the limit: limz→i z z^4 −−i^1.
  1. Find the image of the circle C 1 ( i 2 under the transformation f (x) = (^1) z.
  2. Show that f (z) = 2y − ix is nowhere differentiable.
  3. Differentiate f (z) = iz + i5.
  4. Find the conjugate harmonic function for u(x, y) = 3x^2 y − y^3.