I am having a few issues with Complex Analysis. I've done as much as I could on a few practice problems hoping I would understand it but I'm a bit stuck.

1) Find a parameterization of the line segment joining and . Find the image of the segment under the mapping .

This is what I did:

and

for $\displaystyle =2i(1-t)+(1+t)i$

$\displaystyle =2i-2t+i-1$

$\displaystyle =3i-2it-1$

$\displaystyle f(z)=i(\bar{3i-2it-1})$$\displaystyle =i(\bar{-1+3-2t)i})$

$\displaystyle =i(-1-(3-2t)i)$

$\displaystyle =i(-1-(3i-2it))$

$\displaystyle =i(-1-3i+2it)$

$\displaystyle =-i-3i^{2}+2i^{2}t$

$\displaystyle =-i+3-2t$ for $\displaystyle 0\leq t \leq 1$

The graph doesn't look look right. I've reworked it a few times and I keep getting the same answer. Am I making a silly algebraic mistake somewhere?

2)Find the image $\displaystyle S'$ of the set $\displaystyle S$ under the given complex mapping $\displaystyle w=f(z)$

$\displaystyle f(z)=\bar{z}$; $\displaystyle S$ is the line $\displaystyle y=x$

A friend told me to do it this way: $\displaystyle Re(z)=Im(z)$ $\displaystyle \therefore$ $\displaystyle w=\bar{z}=Re(z)=-Im(z)$

*Is this correct?*

I'll just submit the two problems I'm having the most problem with. Thanks for looking at this.