Maths homework (Laurent series)

OPT287, MTH287 Due: Thu. Feb. 20, at the beginning of class

Homework 3 1. Find and classify the singularities of the following functions:

a)

€

1 (z2 + 4)3

,

b)

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π 1+ z3

,

c)

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(z2 +1)2(z2 −2)−1,

d)

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1 2z2 + 4zi−2

,

e)

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z z −2 + i

,

f)

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z4 − 4 z4 + 4

,

g)

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(z2 + 2)2.

2. Draw the region of convergence of the different series expansions around z = 2 for these functions, specifying the appropriate type of series in each region.

3. Draw the region of convergence of the different series expansions around z = 0 for these functions, specifying the appropriate type of series in each region.

4. Find the form of all series expansions around z = 0 for b, c, e, f, and g.