Proof that index of Jordan curve in the interior is $\pm 1$

Question

These two questions have more or less asked the same thing, but to one no answer is given and to the other one reference to a differential topology book is given and one very sketchy proof is presented. So I'm forced to ask the question again: can someone provide a complex-analytic proof that the index of Jordan curve is $\pm 1$ on the interior of the curve, with index defined in the standard, complex analysis way as a certain integral? A reference to a complex analysis text would suffice, or just the presentation of such a proof. Of course one may, in fact is almost certainly forced to, assume the Jordan curve theorem.