Find a conformal mapping of the half plane $D={z:x >0}$ onto the domain $d^{\prime}={w: |Arg w| < \lambda \pi}$, where $0 <\lambda \le 1$

Question

Find a conformal mapping of the half plane $D=\{z:x >0 \}$ onto the domain $D^{\prime}=\{w: |Arg w| < \lambda \pi \}$, where $0 <\lambda \le 1$.

My work: Say for $\lambda = \frac {1}{4}$, We can map half plane using $f(z)= z^{\frac{1}{2}}$ onto $|Arg w| < \frac {1}{4}\pi $. How can I find general $f(z)= z^{(something)}$ will work for all $0 <\lambda \le 1$? What will be the required conformal map here? I appreciate your kind help. Thank you!

Answer 1

If $a>0$, $z^a$ multiplies angles by $a$. You want to take angles $\pm\pi/2$ to $\pm\lambda\,\pi$, that is, you want $$ a\,\frac\pi2=\lambda\pi\implies a=2\,\lambda. $$