Z/(1+Z)=3+i4

解答

\frac{Z}{1 + Z} = 3 + i 4

Z = - \frac{11}{10} + \frac{1}{5} i

求解步骤

\frac{Z}{1 + Z} = 3 + i 4

替代

Z = x + y i

\frac{x + y i}{1 + x + y i} = 3 + i 4

在两边乘以

1 + x + i y

\frac{x + y i}{1 + x + y i} (1 + x + i y) = 3 (1 + x + i y) + i 4 (1 + x + i y)

乘开

x + i y = (3 + 3 x - 4 y) + i (4 + 3 y + 4 x)

复数仅在实部和虚部均相等时才相等改写为方程组:

[x = 3 + 3 x - 4 y y = 4 + 3 y + 4 x]

[x = 3 + 3 x - 4 y y = 4 + 3 y + 4 x] : x = - \frac{11}{10}, y = \frac{1}{5}

Z = x + y i

代回

Z = - \frac{11}{10} + \frac{1}{5} i

x + 2 y i = (1 + 3 i) (1 + i)

\frac{( 4 + x i )}{2 + i} = y + 2 i

2 x + 7 + (3 - y) i = - 4 + 6 i

a + bi = (a - bi)^{2}

z^{2} - (2 + 2 i) z + (1 + 2 i) = 0