# Simplify a Complex Expression

Suppose that f (z) = x2 - y2 - 2y + iּ(2x - 2xy), where z = x + iy. Use the expressions

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x = (z + z) /2 and y = (z - z)/2i

to write f (z) in terms of z, and simplify the result.

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#### Solution Preview

f (z) = x2 - y2 - 2y + iּ(2x - 2xy) = x^2 - y^2 - 2y + 2ix - 2ixy

z = x + i.y

bar(z) = <z> = x - i.y

x^2 = (z + <z>)^2/4 = (z^2 + <z>^2 + 2.z.<z>)/4

y^2 = (z - <z>)^2/4.i^2 = - (z^2 + ...

#### Solution Summary

A complex expression is simplified. All work is shown.

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