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Show that A is open

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COMPLEX ANALYSIS I Exercise 3, spring 2011

1. Let {z ∈ C| |zz0| > r}. Show that A is open.

2. LetA ={i, 2i, 3i,· · · } ⊂ C.IsAbounded, closed, open? FindA0, A0 and cl(A).

3. Find the line running through the points 1 +i and −3 + 2i a) in a parametric form,

b) in the form ax+by = d, a, b, d ∈ R, c) in the form ¯az +α¯z = γ, α∈ C ja γ ∈ R.

Find also a path joining the points 1 +i,−3 + 2i.

4. Find the limits (if they exist) a) lim

n→∞

in

n, b) lim

n→∞in, c) lim

n→∞

(1 +i)n

n , d) lim

n→∞

2n−in2 (1 +i)n1.

5. Let (an)C be a sequence with lim

n→∞an = a. Show that (an)n=1 is bounded.

6. Show that lim

n→∞ 1 + nzn

=ex(cosy+isiny), when z = x+iy ∈ C.

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