Here's an interesting question I can't seem to get my head around.

A function f is called a diffeomorphism if , where .

Let and denote contours and in a domain D, where .

Let .

A function f is called conformal at a point if , and f is called angle-preserving at a point .

I have shown a function f is conformal at if and only if and is angle-preserving at .

However I am having difficulties proving that if f is conformal at then f is analytic at . I would greatly appreciate any hints, but please don't solve the problem for me.

EDIT: Without using all the theta terms, we wish to show that if the angle between two curves on the z-plane, at is preserved by the mapping and so is the orientation, the f is analytic at