Complex Variables. Complex square root function

  • #1

Homework Statement



Proof that [tex]f(z)=\sqrt{z}=e^{\frac{\ln z}{2}}[/tex] with logarithm branch [tex][0,2\pi)[/tex]. Then [tex]f[/tex] maps horizontal and vertical lines in [tex]A=\mathbb{C}-\{\mathbb{R}^{+}\cup\{0\}\}[/tex] on hyperbola branches.

Homework Equations



I have that [tex]\ln_{[0,2\pi)} (z)=\ln\vert z\vert+i\mathop{\rm arg}\nolimits_{[0,2\pi)} (z)[/tex]

The Attempt at a Solution



I have tried to proof it directly, that means to describe vertical lines with [tex]z(y)=(a,y)[/tex] and horizontal ones with [tex]w(x)=(x,a)[/tex] and substitute in [tex]f[/tex]. That produces an horrific expression that I can't reduce to an hyperbola. What can I do?
 

Answers and Replies

Related Threads on Complex Variables. Complex square root function

  • Last Post
Replies
11
Views
2K
  • Last Post
Replies
6
Views
3K
Replies
3
Views
2K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
7
Views
10K
Replies
5
Views
2K
Replies
1
Views
8K
  • Last Post
Replies
1
Views
2K
Replies
3
Views
3K
Replies
2
Views
1K
Top