Finding the image of a point by analytic continuation of regular branch of $\sqrt{π^2 + \ln^2(z)}$ in $\mathbb{C}\backslash Γ$

Question

Denote $\varphi$ the regular branch of analytic function $\sqrt{π^2 + \ln^2(z)}$ in $\mathbb{C}\backslashΓ$ defined by $φ(1) = \pi$, and where $\Gamma$ is:

With only this information, I would like to find $\varphi(i)$.

But I have no idea how to start because this is, to me, a very hard topic on complex analysis and I'm a bit lost. Any advice what to do, how to understand the problem better or how to start?