Esercizi svolti di analisi reale e complessa - Dipartimento di ...

sin 2 z (0, π
2
( π
2
)
, π) π
1
4
2π
0
dx
a + sin 2 x
1
=
4
= 1
i
|z|=1
|z|=1
dz (iz) −1
a + 1
2i (z − z−1 ) 2 −z dz
z4 − (4a + 2)z2 + 1
± β± β± = (2a+1)± (2a + 1) 2 − 1
α++, α+−, α−+, α−−
• a > 0 β− α−+ α−−
Resα−+ =
α−+
α−−
Resα−− =
(β− − β+)(α−+ − α−−) (β+ − β−)(α−− − α−+) .
=
π
(2a + 1) 2 − 1 .
• a < 0 β+ α++ α+−
Resα++ =
α++ Resα+−
(β+ − β−)(α++ − α+−) =
α++
(β+ − β−)(α+− − α++) .
=
−π
(2a + 1) 2 − 1 .
f(z) =
sgn(a) π
(2a + 1) 2 − 1 .
z 2
z4 +5z2 +6
γR = CR ∪ σR = {|z| = R, Imz ≥ 0} ∪ {−R ≤ x ≤ R} .
R
f(z) z = ±i √ 2 z = ±i √ 3
R