# Thread: complex analysis mapping question

1. ## complex analysis mapping question

(a) Consider the map $\displaystyle f(z)=z^2$. Prove that under $\displaystyle f$, lines parallel to the real axis are mapped to parabolas.
(b) Consider the map $\displaystyle g(z)=\sqrt{z}$, for some branch of the square root. Prove that under $\displaystyle g$ lines parallel to the real axis are mapped to hyperbolas.

I did try to do the questions, so I substitute $\displaystyle z=x+iy$, then get $\displaystyle f(x,y)=x^2+y^2-2ixy$, but I really don't know how to show this mapping thing. Please help me. Thanks a lot.

2. For all $\displaystyle t\in \mathbb{R}$ we have $\displaystyle f(t+bi)^2=t^2+b^2-2bti=u+iv$ then, the transformed curve of the line $\displaystyle L:\; x=t,y=b$ i s

$\displaystyle f (L)\equiv\begin{Bmatrix}u=t^2+b^2\\v=-2bt\end{matrix}$

( parabola if $\displaystyle b\neq 0)$ .

3. Originally Posted by FernandoRevilla
For all $\displaystyle t\in \mathbb{R}$ we have $\displaystyle f(t+bi)^2=t^2+b^2-2bti=u+iv$ then, the transformed curve of the line $\displaystyle L:\; x=t,y=b$ i s

$\displaystyle f (L)\equiv\begin{Bmatrix}u=t^2+b^2\\v=-2bt\end{matrix}$

( parabola if $\displaystyle b\neq 0)$ .
Thank you very much. I did get this, but I just can't continue, because I don't understand why this tells f is parabola? I thought need to show something like v can be written as u^2, then that means parabola. Could you please explain a bit more to me? Thanks a lot for your time.

4. Originally Posted by tsang
Thank you very much. I did get this, but I just can't continue, because I don't understand why this tells f is parabola? I thought need to show something like v can be written as u^2, then that means parabola. Could you please explain a bit more to me? Thanks a lot for your time.

Substitute $\displaystyle t=-v/2b$ in the first equation. What do you obtain?.

5. Part (b) is still quite hard, how do I get it to hyperbolas?

6. Hint

If $\displaystyle \sqrt{t+bi}=x+iy$ then, $\displaystyle 2xy=b$ .

7. Originally Posted by FernandoRevilla
Hint

If $\displaystyle \sqrt{t+bi}=x+iy$ then, $\displaystyle 2xy=b$ .

Thanks for your hint, Professor, I'm do like analysis, but not that good with abstract maths, as I'm only first year doing third year Complex Analysis. I need to keep trying and see if I can get it.

8. Originally Posted by tsang
I need to keep trying and see if I can get it.

Patience.

9. In attempting to do a nearly identical question, I end up with a function:

t=u^2-v^2

or

b=2(t-u^2)(t+v^2)

I can't see any way of eliminating t from this (the non-constant part of the line being mapped)?

10. Originally Posted by Aylix
In attempting to do a nearly identical question, I end up with a function:

t=u^2-v^2

or

b=2(t-u^2)(t+v^2)

I can't see any way of eliminating t from this (the non-constant part of the line being mapped)?
That's true, I get the idea, but can't finish that bit of algebra manipulation.