What is the answer to solvefor z,w= 1/(z+i) ?

The answer to solvefor z,w= 1/(z+i) is z= 1/w-i

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solvefor z,w= 1/(z+i)

solve for

z, w = \frac{1}{z + i}

z = \frac{1}{w} - i

Solution steps

w = \frac{1}{z + i}

Substitute

z = x + y i

w = \frac{1}{x + y i + i}

Multiply both sides by

x + i (1 + y)

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

Expand

w x + i (w + w y) = 1

Rewrite

1

in standard complex form:

1 + 0 i

w x + i (w + w y) = 1 + 0 i

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

[w x = 1 w + w y = 0]

[w x = 1 w + w y = 0] : x = \frac{1}{w}, y = - 1, w \neq = 0

Substitute back

z = x + y i

z = \frac{1}{w} - i

∣ z + 1 + i ∣ = 2

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

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

2 z^{2} + (- 2 - 2 i) z - 6 + 9 i = 0

a + bi - \frac{1}{a + bi} = 4 + 4 i

What is the answer to solvefor z,w= 1/(z+i) ?
The answer to solvefor z,w= 1/(z+i) is z= 1/w-i