How to break $\ln(x+i \cdot y)$ into real and imaginary components? Wolfram shows that there are indeed Re and Im components without stating them: http://www.wolframalpha.com/input/?i=ln(x%2Bi*y)
How do I break $\ln(x+i \cdot y)$ into real and imaginary components?

We want to express $x+iy$ as $re^{i\theta}$, for some value of $r$ and $\theta$. 
First let's normalize $x+iy$ as $\sqrt{x^2+y^2}(\frac{x}{\sqrt{x^2+y^2}}+\frac{y}{\sqrt{x^2+y^2}}i)$.
We see that $r=\sqrt{x^2+y^2}$ and we see that $\tan(\theta)=\frac{y}{x}$. Therefore, we have $\theta=\arctan(\frac{y}{x})$. We can then write $x+iy=\sqrt{x^2+y^2}e^{i\arctan(\frac{y}{x})}$. So we have:
$$\ln(x+iy)=\ln(\sqrt{x^2+y^2}e^{i\arctan(\frac{y}{x})})$$
$$\ln(x+iy)=\frac{1}{2}\ln(x^2+y^2)+i\arctan(\frac{y}{x})$$
Our real part is $\frac{1}{2}\ln(x^2+y^2)$ and our imaginary part is $i\arctan(\frac{y}{x})$.
