Exercises 3.2
21.
(a). For $\omega = sinz$, what is the image of the semi-infinite strip
$S_1 = \{x+iy|-\pi<x<\pi,y>0\}$
(b). what is the image of the smaller semi-infinite strip
$S_2 = \{x+iy|-\frac{\pi}{2}<x<\frac{\pi}{2},y>0\}$
Solutions:
First of all, let‘s assume $z = x + iy$, then expand the $\omega$,
$sin(x+iy)=sinx\cdot coshy+icosx\cdot sinhy$
In addition, observe closely, we will find that it‘s really hard to draw the $w-plane$, whatever the method we use, including "Freeze" Variable and expressing the formula in terms of $\displaystyle e^z$. But now, we can use the concept linear independence on functions to solve the problems!
Namely, if we assume $f=sinx\cdot coshy$,$g=cosx\cdot sinhy$, the value of $g$ doesn‘t affect that of $f$! OR, the other way round.
Proof: let‘s assume $c_1,c_2 \in C$, and $c_1 f+c_2 g = 0$,then
$c_1 tanx \cdot tanhy+c_2=0$
if, $c_1 \ne 0$, we have $\displaystyle tanx\cdot tanhy + \frac{c_2}{c_1}=0$. Since $x, y$ vary freely in the interval, it‘s quite obvious that it‘s impossible for $c_1$ to be $0$.
Thus, $c_1 = 0$, and $c_2 = 0$.
So, to draw the picture of $\omega$, we just need to find the range of $f$ and $g$.
The remaining parts are left for the readers.
原文地址：https://www.cnblogs.com/raymondjiang/p/12233389.html