Logarithmic Mapping Complex Variable and Function Differentiable Questions

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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 Purchase answer to see full attachment Tags: Math Problems Complex variable line segments Logarithmic Mapping Function Differentiable User generated content is uploaded by users for the purposes of learning and should be used following Studypool's honor code & terms of service.

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