 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-1/7/a/solution

Complex differentiability requires the same limit for increments along the real and imaginary directions. Real increments give

f^{'} = u_{x} + i v_{x}

. Imaginary increments give

f^{'} = \frac{u _{y} + i v _{y}}{i} = v_{y} - i u_{y} .

u_{x} = v_{y}, u_{y} = - v_{x} .

They hold throughout the open region where

f

is holomorphic. Conversely, continuous first real partial derivatives satisfying these equations on an open set ensure holomorphy there.

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