First, write , so that

Now we need to check the Cauchy-Riemann Equations hold so that we know we can differentiate the function: and .

So the Cauchy Riemann Equations hold for all x and y, which means the complex function can be differentiated everywhere. We know that maxima and minima occur where the derivative is 0, so

Since the second derivative is positive, the function is minimised when and the minimum value is .