Let f(x, y) congruent u(x, y) + i v(x, y), where z congruent x + i y, so d z = d x + i d y. The total derivative of f with respect to z is then (d f)/(d z) | = | (df)/(dx) (dx)/(dz) + (df)/(dy) (dy)/(dz) | = | 1/2((df)/(dx) - i(df)/(dy)).

analytic function | anti-analytic function | Cauchy integral theorem | complex derivative | conformal mapping | entire function | monogenic function | polygenic function