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hoja de ejercicios de números complejos
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Departamento de Análisis Matemático y Matemática AplicadaGRADO EN INGENIERÍA GEOLÓGICA.
Matemáticas I. Curso 2020-2021
Hoja 1: Los Números Complejos
1 Realiza las siguientes operaciones:
−3 − 3i i−1 1 1
a) b) c) − d) (1 + i)5
4 + 3i i(−i − 1) 1 + 3i 1 − 3i
√
2 Dados los números complejos z1 = 1 − 3i y z2 = 2(cos45o + isen45o ), calcula:
z1 √
a) z 1 · z2 b) c) z 1 8 d) 5 z2
z2
3 Calcula ( √12 + √i )2
2
usando la forma binómica y después usando la forma trigonométrica.
4 Calcula las siguientes potencias:
π π 20
√ 6 √
2
√
2 4
√ 12
1+i 3
a) (cos 12 + isen 12 ) b) (1 − 3i) c) ( 2
− 2
i) d) 1−i
3πi 4πi
5 Sean z1 = 4e 4 ; z2 = 5e 3 . Calcula:
z1
a) z1 · z2 b) c) z16 d) z 1
z2
6 Calcula las siguientes raices:
√
5
√
3
√
4
√
3
√
4
p
3
√
a) −1 b) i c) −i d) i+1 e) 1−i f) −4 + 4 3i
7 Resuelve las siguientes ecuaciones:
a) z 2 + 9 = 0 b) z 2 + 2z + 5 = 0 c) z 3 + z 2 + 4z + 4 = 0 d) z 4 − 7z 2 − 18 = 0
e) z 3 − 2z − 4 = 0 f ) z 2 + iz + 1 = 0 g) iz = 1 + i
8 Factoriza en C los siguientes polinomios:
a) x4 + 1 b) x4 + x2 + 1
9 Describe el conjunto de los números complejos tales que:
a) Imz≤ 3 b) Rez+ Imz=1 c) z · z = 1 d) z = −z
10 Prueba las siguientes igualdades:
a) |z| = |z| b) z = z c) z + w = z + w d) z · w = z · w e) −z = −z
f ) z −1 = z −1
1+z
11 Sea z 6= 1, −1 y con |z| = 1. Prueba que 1−z
es un número imaginario puro.
12 Sea z un número complejo de módulo 1. Prueba que z + z −1 es un número real.
13 Obtener mediante la fórmula de Moivre sen2α; cos2α; sen3α y cos3α.
14 Demostrar que (cosα + isenα)2 = cos2α + isen2α
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