Express each function in the form $u(x,y) + iv (x,y)$ I was doing some homework with complex numbers and I'm stuck with these two, I hope that someone can solve these and clear it up for me. Thank you.


*

*ln(1+z)

*z/(3+z)
Samples,


For the first one a function $w=re^{i\theta+2\pi n}$ satisfies $$\ln w=\ln re^{i\theta}=\ln r+i(\theta+2\pi n)$$Now $w=1+z=1+x+iy$.  So $$r=\sqrt{(1+x)^2+y^2} \ \  \text{and} \ \  \theta=\arctan\{\frac{y}{1+x}\}$$ 
For the second one use $$\frac{1}{u+iv}=\frac{1}{u+iv}\cdot \frac{u-iv}{u-iv}=\frac{u-iv}{u^2+v^2}=\frac{u}{u^2+v^2}-i\frac{v}{u^2+v^2}$$ So,
$$\frac{z}{3+z}=\frac{x+iy}{3+x+iy}
\\ =\frac{x+iy}{3+x+iy} \cdot \frac{3+x-iy}{3+x-iy} 
\\ =\frac{3x+x^2+y^2+i3y}{(3+x)^2+y^2}  
\\ =\frac{3x+x^2+y^2}{(3+x)^2+y^2}+i \frac{3y}{(3+x)^2+y^2}$$
