 Solution

ID: past-exam-of-the-algebra-and-analysis-ii-for-mathematics-and-philosophy-course-of-the-university-of-oxford/2013/4/a/solution

A function is holomorphic on an open set when it is complex differentiable at every point of that set. Explicitly, for every

z \in U

, the limit

f^{'} (z) = h \to 0, h \in C ∖ {0} lim \frac{f ( z + h ) - f ( z )}{h}

exists as a finite complex number. The limit is independent of the direction in which the complex increment approaches zero; existence of derivatives only along the real axis would not suffice.

 New to topics? Read the docs here!