Esercizi sui numeri complessi (1) Determinare la somma dei ...
Esercizi sui numeri complessi (1) Determinare la somma dei ...
Esercizi sui numeri complessi (1) Determinare la somma dei ...
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<strong>Esercizi</strong> <strong>sui</strong> <strong>numeri</strong> <strong>complessi</strong>(1) <strong>Determinare</strong> <strong>la</strong> <strong>somma</strong> <strong>dei</strong> seguenti <strong>numeri</strong> <strong>complessi</strong>:(a) 5 − 3i; −5 + 3i (b) 7 + 2i; 2 − 7i(c) 8i; 5 − 18i (d) 17 − 2i; 3i − 2(e) − 15 + 3i; −5 (f) − 1 − i; 2 + i.(2) <strong>Determinare</strong> il prodotto <strong>dei</strong> seguenti <strong>numeri</strong> <strong>complessi</strong>:(a) 5 − 3i; −5 + 3i (b) 7 + 2i; 2 − 7i(c) 8i; 5 − 18i (d) 17 − 2i; 3i − 2(e) − 15 + 3i; −5 (f) − 1 − i; 2 + i(g) 5 − 3i; 5 + 3i (h) 7 + 2i; 7 + 2i(i) 8i; 8i(l) 8i; −2i(m) − 15 + 3i; i (n)1 − i; −i.(3) <strong>Determinare</strong> il rapporto <strong>dei</strong> seguenti <strong>numeri</strong> <strong>complessi</strong>:(a) 5 − 3i; −5 + 3i(c) 8i; 5 − 18i(e) − 15 + 3i; −5(g) 5 − 3i; 5 + 3i(i) 8i; 1 − 8i(m) − 15 + 3i; i(b) 7 + 2i; 2 − 7i(d) 17 − 2i(f) − 1 − i; 2 + i(h) 7 + 2i; 7 + 2i(l) 8i; −2i(n) − 1 − i; −i.(4) <strong>Determinare</strong> il modulo, il coniugato e l’inverso <strong>dei</strong> seguenti<strong>numeri</strong> <strong>complessi</strong>:(a) 5 − 3i (b) − 5 + 3i (c) 7 + 2i (d) 2 − 7i(e) 8i (f) 5 − 18i (g) 17 − 2i (h) − i(i) − 15 + 3i (l) − 5 − i; (m) − 1 − i (n) 2 + i(o) − 3i (p) √ 3 + √ 6i (q) 7 + 2i (r) i(s) √ 3/2 + i/2 (t) √ 3 − i (u) √ 2 + √ 2i(v) √ 2 − √ 2i (z) − 1.(5) Descrivere l’insieme <strong>dei</strong> <strong>numeri</strong> <strong>complessi</strong> z per cui:(a) |z − (1 − i)| = 2 (b) (1 − i)z − (1 + i)z = i (c) |z| < |z + 1|(d) |z + z| + |z − z| ≤ 2 (e) Iz − |z + z| 2 < 1.(6) Scrivere in forma trigonometrica i seguenti <strong>numeri</strong> <strong>complessi</strong>:(a) 1 − i (b) 1 + √ 3i (c) ρ(cos(θ) − i sin(θ))(d) − 4 (e) − √ 3 − i (f) √ 2 − i √ 2.(7) Verificare che (1 − i) 7 = 8(i + 1).1
2(8) Calco<strong>la</strong>re le seguenti radici n-sime:(a) 4√ 1 − i (b) 2√ ρ(cos(θ) − i sin(θ)) (c) 3√ 1(d) 3√ −1 (e) 3√ i (f) 4√ 1(g) 4√ −1 (h) 4√ 16i (i) 3√ 4 √ 2 − 4 √ 2i(l) 5√ i (m) 5√ −1 (n) √ 1 + 3√ −1(o) √ 1 + √ −1 (p) √ 1 − 3√ −1 (q) √ 1 − √ −1(r) √ −1 + √ −1.(9) Risolvere in campo complesso le equazioni algebriche:(a) z 2 + 5z + 6 = 0 (b) z 2 + z + 1 = 0(c) z 2 + 1 = 0 (d) z 4 + 1 = 0(e) z 4 − 3z 2 + 2 = 0 (f) z 4 + 3z + 2 = 0(g) z 4 − z 3 + z 2 = 0 (h) z 2 − 3iz − 2 = 0(i) z 2 − (1 + i)z + i = 0 (l) z 2 − (2 + i)z + 3i − 3 = 0(m) z 3 − 4iz 2 − 5z + 2i = 0 (n) z 3 − 2z 2 + z − 2 = 0.