# Math Help - exponential

1. ## exponential

how do you solve l 1- e^(ix) l = k l sin (x/2) l for all real x.
fink k.

thanks!

2. Originally Posted by alexandrabel90
how do you solve l 1- e^(ix) l = k l sin (x/2) l for all real x.
fink k.

thanks!
$|1-e^{ix}|=k\left|\sin \frac{x}{2}\right| \Longrightarrow |1-\cos x-i\sin x|^2=k^2\sin^2\frac{x}{2}\Longrightarrow$ $1-2\cos x+1=k^2\sin^2\frac{x}{2}\Longrightarrow 4\sin^2\frac{x}{2}=k^2\sin^2\frac{x}{2}$

$\Longrightarrow\,\, either\,\,\sin \frac{x}{2}=0\Longleftrightarrow \frac{x}{2}=2m\pi\,,\,\,m\in \mathbb{Z} \,,\,\,or\,\,\,k^2=4\Longleftrightarrow k=\pm 2$

We used above $\cos x =\cos^2\frac{x}{2}-\sin^2\frac{x}{2}$ , and also $\forall\,x\,,\,y\in \mathbb{R}\,,\,\,|x+iy|^2=x^2+y^2$

Tonio

3. by the way, how do you get from the 2nd equation to the 3rd equation of (1-2cos x +1)?