Understanding Complex Numbers as Parametric Functions

[Niles]

Homework Statement

Hi all.

I am given the following parametric function in the complex plane C:

$$\gamma = \left\{ {\begin{array}{*{20}c}<br /> {t^2 + it\,\,\,\,\,\,\,\,{\rm{for }}\,\,t \in [0,1]} \\<br /> {t + i\,\,\,\,\,\,\,\,\,\,\,\,{\rm{for }}\,\,t \in ]1,2]} \\<br /> \end{array}} \right.$$

In order to sketch it for t in [0,1], will it be correct it I set x(t) = t² and y(t) = t, and sketch it in the real plane?

Thanks in advance.Niles.

 

[HallsofIvy]

Homework Statement

Hi all.

I am given the following parametric function in the complex plane C:

$$\gamma = \left\{ {\begin{array}{*{20}c}<br /> {t^2 + it\,\,\,\,\,\,\,\,{\rm{for }}\,\,t \in [0,1]} \\<br /> {t + i\,\,\,\,\,\,\,\,\,\,\,\,{\rm{for }}\,\,t \in ]1,2]} \\<br /> \end{array}} \right.$$

In order to sketch it for t in [0,1], will it be correct it I set x(t) = t² and y(t) = t, and sketch it in the real plane?

Thanks in advance.


Niles.

For $0\le t\le 1$, yes. For $1< t\le 2$, x= t, y= 1. Draw those two pieces.

 

[Niles]

Thanks.

Lets look at e.g. w = z² = r²e^i2K = r²(cos(2K) + isin(2K)), where K is the argument of z and r is the modulus. If I wish to plot w = z², then can I do this by plotting x(t) = r²cos(2K) and y(t) = r²sin(2K) as well?Niles.

 

[Niles]

The reason why I am asking is that I seem to get confused when I look at complex numbers as mere parametric functions. Is it correct to look at them in this sense?