[Mathematics][Fundamentals of Complex Analysis][Small Trick] The Trick on drawing the picture of sin(z), for z in Complex Plane

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.