[Mathematics][Fundamentals of Complex Analysis][Small Trick] The Trick on drawing the picture of sin

Posted 2021-03-13 raymondjiang

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 $omega$,

$sin(x+iy)=sinxcdot coshy+icosxcdot sinhy$

　　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 $f=sinxcdot coshy$,$g=cosxcdot sinhy$, 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, $c_1 e 0$, we have $displaystyle tanxcdot tanhy + frac{c_2}{c_1}=0$. 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.