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**manjohn12** Show that for all points $\displaystyle z $ in the right half plane $\displaystyle x>0 $ the function $\displaystyle \text{Log} \ z $ can be written $\displaystyle \text{Log} \ z = \frac{1}{2} \text{Log} \ (x^2+y^2) + i \arctan \frac{y}{x} $.

So $\displaystyle \text{Log} \ z = \text{Log} \ r + i \arg z $. Now $\displaystyle r = |z| = \sqrt{x^2+y^2} $. So $\displaystyle \text{Log} \ z = \frac{1}{2} \text{Log} \ (r^2) + i \arctan \frac{y}{x} = \text{Log} \ r + i \arctan \frac{y}{x}$ where $\displaystyle \arg z = \arctan \frac{y}{x}$.

Is this correct?