What is the answer to z-1/z =1+3i ?

The answer to z-1/z =1+3i is z=-0.13355…+0.31619…i,z=1.13355…+2.68380…i

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z-1/z =1+3i

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

z = - 0.13355 \dots + 0.31619 \dots i, z = 1.13355 \dots + 2.68380 \dots i

Solution steps

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

Substitute

z = x + y i

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

Multiply both sides by

x + y i

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

Expand

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

Complex numbers can be equal only if their real and imaginary parts are equalRewrite as system of equations:

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

[x^{2} - y^{2} - 1 = x - 3 y 2 x y = y + 3 x] : (x = - 0.13355 \dots, x = 1.13355 \dots, y = 0.31619 \dots y = 2.68380 \dots)

Substitute back

z = x + y i

z = - 0.13355 \dots + 0.31619 \dots i, z = 1.13355 \dots + 2.68380 \dots i

z^{2} + i z = 2 i^{2}

a + ib = (4 + 3 i) (5 - 2 i)

\frac{- 1 + 2 i}{8 i} = a + bi

z^{2} - 8 z + 18 = 8 i

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

What is the answer to z-1/z =1+3i ?
The answer to z-1/z =1+3i is z=-0.13355…+0.31619…i,z=1.13355…+2.68380…i