In my book on complex analysis a "Holomorphically simply connected" set is defined as a set where for any holomorphic function $f $ and any closed path $\gamma _1 $ we have that $\int_{\gamma _1 } f(z) dz =0 $
My question is:
Given two closed paths $\gamma _1 $ and $\gamma _2 $ in a holomorphically simply connected set, how can i construct a path homotopy between them, and also show that they are path homotopic to the constant path, so that the set is simply connected?
Thanks in advance!