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?