Express each function in the form $u(x,y) + iv (x,y)$

Question

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.

1. ln(1+z)

2. z/(3+z)

Samples,

Answer 1

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, $$\frac{z}{3+z}=\frac{x+iy}{3+x+iy} \\ =\frac{x+iy}{3+x+iy} \cdot \frac{3+x-iy}{3+x-iy} \\ =\frac{3x+x^2+y^2+i3y}{(3+x)^2+y^2} \\ =\frac{3x+x^2+y^2}{(3+x)^2+y^2}+i \frac{3y}{(3+x)^2+y^2}$$

Answer 2

For the second, note that $$ \frac{z}{3+z} = (x + iy)(3 + x + iy)^{-1} $$ and that $$ (a + ib)^{-1} = \frac{a - ib}{a^2 + b^2} $$

Does that help?