What is the answer to z-(1/z)=1+5i ?

The answer to z-(1/z)=1+5i is z=-0.04321…+0.19889…i,z=1.04321…+4.80110…i

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z-(1/z)=1+5i

z - (\frac{1}{z}) = 1 + 5 i

z = - 0.04321 \dots + 0.19889 \dots i, z = 1.04321 \dots + 4.80110 \dots i

Solution steps

z - (\frac{1}{z}) = 1 + 5 i

Substitute

z = x + y i

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

Expand

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

[x^{2} - y^{2} - 1 = x - 5 y 2 x y = y + 5 x] : (x = - 0.04321 \dots, x = 1.04321 \dots, y = 0.19889 \dots y = 4.80110 \dots)

Substitute back

z = x + y i

z = - 0.04321 \dots + 0.19889 \dots i, z = 1.04321 \dots + 4.80110 \dots i

z^{2} = - (\frac{4 i}{3})

\frac{1 + i}{z} - (4 + 3 i) = i

z^{2} - (5 + 2 i) z + 1 = 0

+ 6 i z^{2} = 3 + 5 i, z^{1} = 2

(2 + 3 i) (a + bi) = 13

What is the answer to z-(1/z)=1+5i ?
The answer to z-(1/z)=1+5i is z=-0.04321…+0.19889…i,z=1.04321…+4.80110…i