in {|z| < 1}, {1

Analisi Complessa– a.a. 10/11
Homework 21
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2.1. (i) Calculate the Laurent expansion about z0 = 0 of f (z) = (z−i)(z+2i)
in {|z| < 1}, {1 < |z| < 2}
and in {|z| > 2}.
z
about z1 = −i, about z2 = 3 and
(ii) Calculate the Laurent expansion of g(z) = (z+i)(z−3)(z−5i)
about z3 = 5i.
2.2. Show that the set of the one-to-one and onto holomorphic maps of C to itself (a group under
composition) is
ϕ : ϕ(z) = az + b, where a ∈ C \ {0}, b ∈ C .
2.3. Compute the residue fo the given function at the assigned point:
z2
(i) f (z) = (z−2i)(z+3)
, z0 = 2i;
(ii) f (z) =
(iii) f (z) =
(iv) f (z) =
z 2 +1
, z0 = −3;
z(z+3)2
ez
, z0 = 1 + i;
(z−i−1)3
sin z
, z0 = 0
z 3 (z−2)(z+1)
2.4. Calculate
Z the following integrals:
z
(i)
dz;
2 )(z + i)
(9
−
z
Z|z|=2
cos z
(ii)
dz,
2
γ z(z + 8)
where γ = ∂Q and Q is the square center at the origin and side length 4, oriented
counter-clockwise.
2.5. Calculate the following integrals:
Z 2π
1
dθ, where 0 ≤ a < 1;
(i)
1 + a cos θ
0
Z 2π
1
(ii) 2
sin2n θ dθ;
Z
(iii)
0
1
0
π/2
√
[2π/ 1 − a2 ]
2n
−2n
[π2
]
n
p
[π/2 a(a + 1)]
dx
, where a > 0.
a + sin2 x
Homework due on Thursday Dec. 9.
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