# help with this map

• Nov 16th 2006, 04:21 AM
splash
help with this map
Show that the mapping w = sin z maps the y-axis one-to-one and onto the v-axis.
• Nov 16th 2006, 06:18 AM
ThePerfectHacker
Quote:

Originally Posted by splash
Show that the mapping w = sin z maps the y-axis one-to-one and onto the v-axis.

I think you want to show it maps the y-axis one-to-one and onto the x-axis.

The y-axis are the imaginary numbers $\displaystyle xi$.

But that is not true,
$\displaystyle \sin (xi)=xi+\frac{x^3i}{3!}+\frac{x^5i}{5!}+\frac{x^7i }{7!}+...=i\sinh x$
It still remains an imaginary number. So you established no map.
• Nov 16th 2006, 07:54 AM
CaptainBlack
Quote:

Originally Posted by ThePerfectHacker
I think you want to show it maps the y-axis one-to-one and onto the x-axis.

The y-axis are the imaginary numbers $\displaystyle xi$.

But that is not true,
$\displaystyle \sin (xi)=xi+\frac{x^3i}{3!}+\frac{x^5i}{5!}+\frac{x^7i }{7!}+...=i\sinh x$
It still remains an imaginary number. So you established no map.

Where $\displaystyle w=u+\bold{i}v$, and $\displaystyle z=x+\bold{i}y$,
so now it does.

RonL