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1. Perform the required calculation and express the answer(s) in the form a + bi

(a) (2+3i)/1 – 2i – (8 + i)/(6 – i)

(b) (1 + i)100

(c) (3 + 4i)1/2

(d) log(1 – i)

(e) i

2. Describe or sketch the sets of points determined by the following relations

(a) Re((1 + i)z – 1) = 0

(b) 121 = 12 – il

3. (a) Find the image of the circle [2] = 4 under the logarithmic mapping w = Logz.

(b) Is the function u(r,y) = 4.ry3 – 42’y+harmonic? if it is, state the domain where

it’s harmonic and find its harmonic conjugate v(x,y).

4. (a) Is the following function differentiable? is it analytic? if yes, determine its domain

of analyticity and its derivative.

f(x) = 4.rº +5.2 – 4y2 +9+il8ry + 5y – 1)

(b) Evaluate Sr (22 + 4)dz, where I’ is the line segment from z = to z=1+i.

1

5. (a) Does the function f(x) = has a Maclaurin series (i.e., Taylor series cen-

(1 +22)

tered at 20 = 0). If it does, find the series and the domain where this series

converges.

1

(b) Check if the function f(2)=

has a Laurent series expansion in the

(2-1)(2-3)

domain 0 < 12 – 11 < 2. If it does, find its Laurent series.
6. (a) Determine whether 2 = 0) is an essential singularity of f(x) = (2+1/3.
e*
(b) Using the residue theorem, find Sc 24 – 3i23 – 222
dz, where is the circle 2= 4
traversed once with counterclockwise orientation.
(c) (Extra credit, 1 point)
Let f(2) have an isolated singularity at zo and suppose that f(2) is bounded in
some punctured neighborhood of zo (i.e., 0< 12 – Zo
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Tags:
Math Problems
Complex variable
line segments
Logarithmic Mapping
Function Differentiable
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