Conformal Mapping: Finding \phi(z) = z^{0.5}

[Niles]

Homework Statement

Hi

Is there a rigorous way to find conformal mappins? Say I would like to find how $$\phi(z)=z^{0.5}$$ maps the domain $r\exp(i\phi)$ (with $r>0$ and $0\leq \phi \leq \pi$), how would I do this?

Thanks in advance.

 

[pasmith]

Homework Statement

Hi

Is there a rigorous way to find conformal mappins? Say I would like to find how $$\phi(z)=z^{0.5}$$ maps the domain $r\exp(i\phi)$ (with $r>0$ and $0\leq \phi \leq \pi$), how would I do this?

Please don't use the same symbol for two different objects in the same context! Either $\phi$ is a complex function or it's the argument of a complex number. Choose one and stick with it, and find a different symbol for the other.

To answer your question: Start with $\phi(re^{i\theta}) = r^{1/2}e^{i\theta/2}$. What values can $r^{1/2}$ take if $r > 0$? What values can $\theta/2$ take if $0 \leq \theta \leq \pi$? What region of the complex plane does that give you?