What is the answer to z=z+i/z ?

The answer to z=z+i/z is No Solution

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z=z+i/z

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

No Solution

Solution steps

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

Substitute

z = x + y i

x + y i = x + y i + \frac{i}{x + y i}

Multiply both sides by

x + y i

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

Expand

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

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

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

[x^{2} - y^{2} = x^{2} - y^{2} 2 x y = 1 + 2 x y] : No Solution

Substitute back

z = x + y i

No Solution

s i (x + 3)^{2} = (x - 1)^{2} + 18

x, f s i f (x) = x^{2} - 3 x + 2

x^{2} + 2 - i = 0

x + y i - x i + y = 3

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