Question

Re z = 5, find the maximum value of the expression: Re(z^2) - Im(z^2)

Answer 1

Let \(z = x + iy\), \(Re(z) = x\)  \((\therefore x = 5)\) 

\(z^2 = (x + iy)^2\)

\(= x^2 - y^2 + i2xy\)

\(Re (z^2) = x^2 - y^2 = 5-y^2\)    \((\because x = 5)\)

\(Im (z^2) = 2xy = 10 y\)

\(Re(z^2) - Im(z^2) = 5 - y^2 - 10y\)

\(= 5 - (y^2 + 10y + 25) +25\)

\(= 30 - (y + 5)^2\)

\(\therefore \) Maximum value of \(Re(z^2) - Im(z^2) = 30\)