Analiza I

1.7. KOMPLEKSNA ˇ STEVILA C 27
4.
5.
|z| = (Rez) 2 +(Imz) 2 ≥ (Rez) 2 +0 = |Re(z)|
|z| = (Rez) 2 +(Imz) 2 ≥ 0+(Imz) 2 = |Im(z)|
Od tod in točke 4. sledi
Posledica 6
(∗) Rez =
z +z
2
(∗∗) w·z = w·z = w·z
|z +w| 2 = (z +w)·(z +w)
= (z +w)·(z +w)
= z ·z +w·z +w ·z +w·w
(∗)
= |z| 2 +|w| 2 +2Re(w ·z)
w·z +w·z
Re(wz) =
2
(∗∗) w·z +w ·z
=
2
|z|+|w| 2 = |z| 2 +2|z| |w|+|w| 2
= |z| 2 +2|z| |w|+|w| 2
= |z| 2 +|w| 2 +2|w·z|
|z +w| ≤ |z|+|w|.
|z +w| ≥ |z|−|w|
|z −w| ≥ |z|−|w|