z-1/z =5+3i

解答

z - \frac{1}{z} = 5 + 3 i

z = - 0.14706 \dots + 0.08333 \dots i, z = 5.14706 \dots + 2.91666 \dots i

求解步骤

z - \frac{1}{z} = 5 + 3 i

替代

z = x + y i

x + y i - \frac{1}{x + y i} = 5 + 3 i

在两边乘以

x + y i

x (x + y i) + y i (x + y i) - \frac{1}{x + y i} (x + y i) = 5 (x + y i) + 3 i (x + y i)

乘开

(x^{2} - y^{2} - 1) + 2 i x y = (5 x - 3 y) + i (3 x + 5 y)

复数仅在实部和虚部均相等时才相等改写为方程组:

[x^{2} - y^{2} - 1 = 5 x - 3 y 2 x y = 3 x + 5 y]

[x^{2} - y^{2} - 1 = 5 x - 3 y 2 x y = 3 x + 5 y] : (x = - 0.14706 \dots, x = 5.14706 \dots, y = 0.08333 \dots y = 2.91666 \dots)

z = x + y i

代回

z = - 0.14706 \dots + 0.08333 \dots i, z = 5.14706 \dots + 2.91666 \dots i

(a - bi) (a + bi + 1) = 11 + 3 i

a (1 - 3 i) + b (4 - i) = - 17 - 4 i

(a + ib)^{2} = i

x - 2 y i = - 1 + 5 i

\frac{2 + i}{z} = \frac{2 + 2 i}{z + i}